This is GameTheory.info, produced by Macaulay2, version 1.25.05

GameTheory -- a package for computing equilibria in game theory
***************************************************************

Description
===========

Game Theory is a package for several equilibrium concepts in game theory. It
constructs the algebro-geometric and combinatorial models for Nash, correlated,
dependency, and conditional independence equilibria. The latter three notions
of equilibria are all generalizations of Nash equilibria. An $n$-player game in
normal form is defined by $n$-tensors of format $d_1 \times d_2 \times \cdots
\times d_n$, where $d_i$ is the number of pure strategies of player $i$. The
entry $\{j_1, j_2, \cdots, j_n\}$ of the $i$-th (payoff) tensor for player $i$
is the payoff when player $1$ chooses strategy $j_1$, player $2$ chooses
strategy $j_2$, and so on. One can define a specific game or a random game,
e.g., a list with random tensors.

+----------------------------------------------------------------------------+
|i1 : A = zeroTensor {2,2};                                                  |
+----------------------------------------------------------------------------+
|i2 : B = zeroTensor {2,2};                                                  |
+----------------------------------------------------------------------------+
|i3 : A#{0,0} = 3;  A#{0,1} = 0;  A#{1,0} = 0;  A#{1,1} = 2;                 |
+----------------------------------------------------------------------------+
|i7 : B#{0,0} = 2;  B#{0,1} = 0;  B#{1,0} = 0;  B#{1,1} = 3;                 |
+----------------------------------------------------------------------------+
|i11 : X = {randomTensor {2,2,2}, randomTensor {2,2,2}, randomTensor {2,2,2}}|
|                                                                            |
|o11 = {Tensor{...11...}, Tensor{...11...}, Tensor{...11...}}                |
|                                                                            |
|o11 : List                                                                  |
+----------------------------------------------------------------------------+
|i12 : randomGame {2,2,2}                                                    |
|                                                                            |
|o12 = {Tensor{...11...}, Tensor{...11...}, Tensor{...11...}}                |
|                                                                            |
|o12 : List                                                                  |
+----------------------------------------------------------------------------+

The notion of Nash equilibria is one of the central topics in game theory.
See nashEquilibriumIdeal, computes a
square system of $d_1 + \cdots + d_n$ polynomials that algebraically model the
set of totally mixed Nash equilibria. In the case where the dimension of this
ideal is zero (commonly referred to as a generic game), one can use the mixed
volume of the Newton polytopes of each polynomial in the system to obtain an
upper bound on the number of totally mixed Nash equilibria. The list of these
Newton polytopes is provided by deltaList. The
mixed volume of these polytopes equals the number of certain block
derangements. The method numberTMNE, is typically
faster than mixedVolume (Polyhedra), in this case.

+-----------------------------------------------------------------------------+
|i13 : NR = nashEquilibriumRing X;                                            |
+-----------------------------------------------------------------------------+
|i14 : I = nashEquilibriumIdeal(NR,X)                                         |
|                                                                             |
|               67                  1                 31                      |
|o14 = ideal (- --p      p       - --p      p       - --p      p       +      |
|               18 {1, 0} {2, 0}   20 {1, 1} {2, 0}   20 {1, 0} {2, 1}        |
|      -----------------------------------------------------------------------|
|       7                  43                 19                 39           |
|      --p      p      , - --p      p       - --p      p       + --p      p   |
|      12 {1, 1} {2, 1}     7 {0, 0} {2, 0}   20 {0, 1} {2, 0}    7 {0, 0} {2,|
|      -----------------------------------------------------------------------|
|           13                  35                  1                 17      |
|         + --p      p      , - --p      p       - --p      p       - --p     |
|      1}   63 {0, 1} {2, 1}     9 {0, 0} {1, 0}   24 {0, 1} {1, 0}    2 {0,  |
|      -----------------------------------------------------------------------|
|                  23                                                         |
|        p       - --p      p      , p       + p       - 1, p       + p       |
|      0} {1, 1}    5 {0, 1} {1, 1}   {0, 0}    {0, 1}       {1, 0}    {1, 1} |
|      -----------------------------------------------------------------------|
|      - 1, p       + p       - 1)                                            |
|            {2, 0}    {2, 1}                                                 |
|                                                                             |
|o14 : Ideal of NR                                                            |
+-----------------------------------------------------------------------------+
|i15 : dim I                                                                  |
|                                                                             |
|o15 = 0                                                                      |
+-----------------------------------------------------------------------------+
|i16 : degree I                                                               |
|                                                                             |
|o16 = 2                                                                      |
+-----------------------------------------------------------------------------+
|i17 : D = deltaList {2,2,2}                                                  |
|                                                                             |
|o17 = {Polyhedron{...1...}, Polyhedron{...1...}, Polyhedron{...1...}}        |
|                                                                             |
|o17 : List                                                                   |
+-----------------------------------------------------------------------------+
|i18 : mixedVolume D                                                          |
|                                                                             |
|o18 = 2                                                                      |
|                                                                             |
|o18 : QQ                                                                     |
+-----------------------------------------------------------------------------+
|i19 : blockDerangements {2,2,2}                                              |
|                                                                             |
|o19 = {{set {(2, 0)}, set {(0, 0)}, set {(1, 0)}}, {set {(1, 0)}, set {(2,   |
|      -----------------------------------------------------------------------|
|      0)}, set {(0, 0)}}}                                                    |
|                                                                             |
|o19 : List                                                                   |
+-----------------------------------------------------------------------------+
|i20 : numberTMNE {2,2,2}                                                     |
|                                                                             |
|o20 = 2                                                                      |
+-----------------------------------------------------------------------------+

The set of correlated equilibria of a game forms a convex polytope inside the
(probability) simplex which is the standard simplex of dimension $d_1 \cdots
d_n -1$. Thus, the variables are taken from probabilityRing.
In particular, the map from nashEquilibriumRing,
to probabilityRing, is
the Segre embedding.

+--------------------------------------------------+
|i21 : CE1 = correlatedEquilibria {A, B}           |
|                                                  |
|o21 = CE1                                         |
|                                                  |
|o21 : Polyhedron                                  |
+--------------------------------------------------+
|i22 : dim CE1                                     |
|                                                  |
|o22 = 3                                           |
+--------------------------------------------------+
|i23 : vertices CE1                                |
|                                                  |
|o23 = | 1 0 2/7 3/8 6/25 |                        |
|      | 0 0 3/7 0   9/25 |                        |
|      | 0 0 0   1/4 4/25 |                        |
|      | 0 1 2/7 3/8 6/25 |                        |
|                                                  |
|               4       5                          |
|o23 : Matrix QQ  <-- QQ                           |
+--------------------------------------------------+
|i24 : facets CE1                                  |
|                                                  |
|o24 = (| 0  -1 0  0  |, 0)                        |
|       | -3 2  0  0  |                            |
|       | 0  0  -1 0  |                            |
|       | -2 0  3  0  |                            |
|       | 0  2  0  -3 |                            |
|       | 0  0  3  -2 |                            |
|                                                  |
|o24 : Sequence                                    |
+--------------------------------------------------+
|i25 : CE2 = correlatedEquilibria randomGame{2,2,2}|
|                                                  |
|o25 = CE2                                         |
|                                                  |
|o25 : Polyhedron                                  |
+--------------------------------------------------+
|i26 : dim CE2                                     |
|                                                  |
|o26 = 7                                           |
+--------------------------------------------------+

The algebro-geometric model of dependency equilibria is called Spohn variety.
Its defining ideal is given by rank one conditions on Spohn matrices. One can
also define Konstanz matrices which is crucial to understand the projection of
dependency equilibria to the payoff region.

+-----------------------------------------------------------------------------+
|i27 : PR = probabilityRing {2,2,2};                                          |
+-----------------------------------------------------------------------------+
|i28 : X = randomGame {2,2,2};                                                |
+-----------------------------------------------------------------------------+
|i29 : spohnMatrices(PR,X)                                                    |
|                                                                             |
|o29 = {| p_{0, 0, 0}+p_{0, 0, 1}+p_{0, 1, 0}+p_{0, 1, 1} 5/3p_{0, 0,         |
|       | p_{1, 0, 0}+p_{1, 0, 1}+p_{1, 1, 0}+p_{1, 1, 1} 6/5p_{1, 0,         |
|      -----------------------------------------------------------------------|
|      0}+3p_{0, 0, 1}+8/7p_{0, 1, 0}+6/5p_{0, 1, 1}  |, | p_{0, 0, 0}+p_{0,  |
|      0}+6/7p_{1, 0, 1}+6p_{1, 1, 0}+3/10p_{1, 1, 1} |  | p_{0, 1, 0}+p_{0,  |
|      -----------------------------------------------------------------------|
|      0, 1}+p_{1, 0, 0}+p_{1, 0, 1}                                          |
|      1, 1}+p_{1, 1, 0}+p_{1, 1, 1}                                          |
|      -----------------------------------------------------------------------|
|      9/4p_{0, 0, 0}+1/3p_{0, 0, 1}+1/2p_{1, 0, 0}+1/3p_{1, 0, 1}  |, | p_{0,|
|      3/10p_{0, 1, 0}+3/8p_{0, 1, 1}+1/2p_{1, 1, 0}+8/7p_{1, 1, 1} |  | p_{0,|
|      -----------------------------------------------------------------------|
|      0, 0}+p_{0, 1, 0}+p_{1, 0, 0}+p_{1, 1, 0} 5/7p_{0, 0,                  |
|      0, 1}+p_{0, 1, 1}+p_{1, 0, 1}+p_{1, 1, 1} 3/2p_{0, 0,                  |
|      -----------------------------------------------------------------------|
|      0}+7p_{0, 1, 0}+6/5p_{1, 0, 0}+1/5p_{1, 1, 0}   |}                     |
|      1}+5/3p_{0, 1, 1}+3/2p_{1, 0, 1}+5/4p_{1, 1, 1} |                      |
|                                                                             |
|o29 : List                                                                   |
+-----------------------------------------------------------------------------+
|i30 : spohnIdeal(PR,X);                                                      |
|                                                                             |
|o30 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i31 : konstanzMatrix(PR,X)                                                   |
|                                                                             |
|o31 = | k_0-5/3 k_0-3   k_0-8/7  k_0-6/5 0       0       0       0        |  |
|      | 0       0       0        0       k_0-6/5 k_0-6/7 k_0-6   k_0-3/10 |  |
|      | k_1-9/4 k_1-1/3 0        0       k_1-1/2 k_1-1/3 0       0        |  |
|      | 0       0       k_1-3/10 k_1-3/8 0       0       k_1-1/2 k_1-8/7  |  |
|      | k_2-5/7 0       k_2-7    0       k_2-6/5 0       k_2-1/5 0        |  |
|      | 0       k_2-3/2 0        k_2-5/3 0       k_2-3/2 0       k_2-5/4  |  |
|                                                                             |
|                         6                 8                                 |
|o31 : Matrix (PR[k ..k ])  <-- (PR[k ..k ])                                  |
|                  0   2             0   2                                    |
+-----------------------------------------------------------------------------+

The algebro-geometric model of conditional independence equilibria is obtained
by intersecting the Spohn variety with the conditional independence model of a
given set of conditional independence statements, followed by the removal of
certain components. The set of conditional independence statements can be
generated via graphical models.

+-----------------------------------------------------------------------------+
|i32 : G1 = graph ({}, Singletons => {1,2,3});                                |
+-----------------------------------------------------------------------------+
|i33 : G2 = graph ({{1,2}}, Singletons => {3});                               |
+-----------------------------------------------------------------------------+
|i34 : I1 = spohnCI(PR,X,G1)                                                  |
|                                                                             |
|                                                                             |
|o34 = ideal (727710p          - 2870478p          + 1936669p         ,       |
|                    {1, 0, 1}           {1, 1, 0}           {1, 1, 1}        |
|      -----------------------------------------------------------------------|
|                                                                             |
|      1188593p          + 9157515p          - 12232731p         ,            |
|              {1, 0, 0}           {1, 1, 0}            {1, 1, 1}             |
|      -----------------------------------------------------------------------|
|                                                                             |
|      437292065p          - 70326711p          - 91417042p         ,         |
|                {0, 1, 1}            {1, 1, 0}            {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|                                                                             |
|      24988118p          - 85869p          - 10485198p         ,             |
|               {0, 1, 0}         {1, 1, 0}            {1, 1, 1}              |
|      -----------------------------------------------------------------------|
|                                                                             |
|      22730129187225p          + 9420433731891p          - 24975873351298p   |
|                     {0, 0, 1}                 {1, 1, 0}                  {1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           , 21214787241410p          - 34552109032893p          +           |
|      1, 1}                 {0, 0, 0}                  {1, 1, 0}             |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      22940456357154p         , 140653422p          + 179828669p         p   |
|                     {1, 1, 1}            {1, 1, 0}             {1, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|                        2                                                    |
|            - 366981930p         )                                           |
|      1, 1}             {1, 1, 1}                                            |
|                                                                             |
|o34 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i35 : I2 = spohnCI(PR,X,G2)                                                  |
|                                                                             |
|                                                                             |
|o35 = ideal (330p          - 2240p          + 126p          + 441p         , |
|                 {0, 0, 1}        {0, 1, 1}       {1, 0, 1}       {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      330p          - 2240p          + 126p          + 441p         , p      |
|          {0, 0, 0}        {0, 1, 0}       {1, 0, 0}       {1, 1, 0}   {1, 0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - p         p         , p         p          - p          |
|      1} {1, 1, 0}    {1, 0, 0} {1, 1, 1}   {0, 1, 1} {1, 1, 0}    {0, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         , 215560p         p          + 1031760p         p          |
|      0} {1, 1, 1}         {0, 1, 0} {1, 0, 1}           {0, 1, 1} {1, 0, 1} |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      - 12348p         p          - 56700p          - 2375000p         p     |
|              {1, 0, 0} {1, 0, 1}         {1, 0, 1}           {0, 1, 0} {1,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|            + 1332450p         p          + 71442p         p          -      |
|      1, 1}           {0, 1, 1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}        |
|      -----------------------------------------------------------------------|
|                                                                       2     |
|      269892p         p          + 401310p         p          - 250047p      |
|             {1, 0, 1} {1, 1, 1}          {1, 1, 0} {1, 1, 1}          {1, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        , 215560p         p          + 1031760p         p          -         |
|      1}         {0, 1, 1} {1, 0, 0}           {0, 1, 1} {1, 0, 1}           |
|      -----------------------------------------------------------------------|
|                                        2                                    |
|      12348p         p          - 56700p          - 2375000p         p       |
|            {1, 0, 0} {1, 0, 1}         {1, 0, 1}           {0, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|         + 1332450p         p          + 71442p         p          -         |
|      1}           {0, 1, 1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}           |
|      -----------------------------------------------------------------------|
|                                                                       2     |
|      269892p         p          + 401310p         p          - 250047p      |
|             {1, 0, 1} {1, 1, 1}          {1, 1, 0} {1, 1, 1}          {1, 1,|
|      -----------------------------------------------------------------------|
|                                                    2                        |
|        , 1161652840p         p          - 66543372p          -              |
|      1}             {0, 1, 0} {1, 0, 0}            {1, 0, 0}                |
|      -----------------------------------------------------------------------|
|                                                                             |
|      26613217440p         p          + 12948012p         p          +       |
|                  {0, 1, 1} {1, 0, 1}            {1, 0, 0} {1, 0, 1}         |
|      -----------------------------------------------------------------------|
|                 2                                                           |
|      1462519800p          - 12798875000p         p          + 385000938p    |
|                 {1, 0, 1}               {0, 1, 0} {1, 1, 0}             {1, |
|      -----------------------------------------------------------------------|
|                                   2                                         |
|           p          + 2162659590p          + 68441323050p         p        |
|      0, 0} {1, 1, 0}              {1, 1, 0}               {0, 1, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|         - 34369215300p         p          - 3297222936p         p          +|
|      1}               {0, 1, 1} {1, 1, 1}              {1, 0, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      6961594248p         p          - 11698893423p         p          +     |
|                 {1, 0, 1} {1, 1, 1}               {1, 1, 0} {1, 1, 1}       |
|      -----------------------------------------------------------------------|
|                 2                                                        2  |
|      6449712318p         , 197728876800p         p          - 4224976000p   |
|                 {1, 1, 1}               {0, 1, 0} {0, 1, 1}              {0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|            + 38550752520p         p          - 465976476p         p         |
|      1, 1}               {0, 1, 1} {1, 0, 1}             {1, 0, 0} {1, 0, 1}|
|      -----------------------------------------------------------------------|
|                   2                                                         |
|      - 2139687900p          + 48895744380p         p          -             |
|                   {1, 0, 1}               {0, 1, 0} {1, 1, 1}               |
|      -----------------------------------------------------------------------|
|                                                                             |
|      30970790200p         p          - 7285499046p         p          -     |
|                  {0, 1, 1} {1, 1, 1}              {1, 0, 0} {1, 1, 1}       |
|      -----------------------------------------------------------------------|
|                                                                             |
|      17660535204p         p          - 19791034830p         p          +    |
|                  {1, 0, 1} {1, 1, 1}               {1, 1, 0} {1, 1, 1}      |
|      -----------------------------------------------------------------------|
|                 2                             2                             |
|      6724509561p         , 249341254595596800p          -                   |
|                 {1, 1, 1}                     {0, 1, 0}                     |
|      -----------------------------------------------------------------------|
|                      2                           2                          |
|      113841978320000p          - 587608451624376p          -                |
|                      {0, 1, 1}                   {1, 0, 0}                  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      231645727830554520p         p          + 73986638952996p         p     |
|                         {0, 1, 1} {1, 0, 1}                  {1, 0, 0} {1,  |
|      -----------------------------------------------------------------------|
|                                2                                            |
|            + 12729437966655900p          + 61658804952533880p         p     |
|      0, 1}                     {1, 0, 1}                     {0, 1, 0} {1,  |
|      -----------------------------------------------------------------------|
|                                                                       2     |
|            - 9187203719981196p         p          - 24957009487535580p      |
|      1, 0}                    {1, 0, 0} {1, 1, 0}                     {1, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|         + 497877041974323900p         p          - 301331166584291900p      |
|      0}                      {0, 1, 0} {1, 1, 1}                      {0, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 38578437577926738p         p          +                 |
|      1} {1, 1, 1}                     {1, 0, 0} {1, 1, 1}                   |
|      -----------------------------------------------------------------------|
|                                                                             |
|      60390694118826084p         p          - 82557682628561784p         p   |
|                        {1, 0, 1} {1, 1, 1}                     {1, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|                                2                                            |
|            + 56572267000026519p         )                                   |
|      1, 1}                     {1, 1, 1}                                    |
|                                                                             |
|o35 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i36 : J = probabilitySumIdeal(PR)                                            |
|                                                                             |
|o36 = ideal(p          + p          + p          + p          + p          + |
|             {0, 0, 0}    {0, 0, 1}    {0, 1, 0}    {0, 1, 1}    {1, 0, 0}   |
|      -----------------------------------------------------------------------|
|      p          + p          + p          - 1)                              |
|       {1, 0, 1}    {1, 1, 0}    {1, 1, 1}                                   |
|                                                                             |
|o36 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i37 : J1 = I1 + J                                                            |
|                                                                             |
|                                                                             |
|o37 = ideal (727710p          - 2870478p          + 1936669p         ,       |
|                    {1, 0, 1}           {1, 1, 0}           {1, 1, 1}        |
|      -----------------------------------------------------------------------|
|                                                                             |
|      1188593p          + 9157515p          - 12232731p         ,            |
|              {1, 0, 0}           {1, 1, 0}            {1, 1, 1}             |
|      -----------------------------------------------------------------------|
|                                                                             |
|      437292065p          - 70326711p          - 91417042p         ,         |
|                {0, 1, 1}            {1, 1, 0}            {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|                                                                             |
|      24988118p          - 85869p          - 10485198p         ,             |
|               {0, 1, 0}         {1, 1, 0}            {1, 1, 1}              |
|      -----------------------------------------------------------------------|
|                                                                             |
|      22730129187225p          + 9420433731891p          - 24975873351298p   |
|                     {0, 0, 1}                 {1, 1, 0}                  {1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           , 21214787241410p          - 34552109032893p          +           |
|      1, 1}                 {0, 0, 0}                  {1, 1, 0}             |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      22940456357154p         , 140653422p          + 179828669p         p   |
|                     {1, 1, 1}            {1, 1, 0}             {1, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|                        2                                                    |
|            - 366981930p         , p          + p          + p          +    |
|      1, 1}             {1, 1, 1}   {0, 0, 0}    {0, 0, 1}    {0, 1, 0}      |
|      -----------------------------------------------------------------------|
|      p          + p          + p          + p          + p          - 1)    |
|       {0, 1, 1}    {1, 0, 0}    {1, 0, 1}    {1, 1, 0}    {1, 1, 1}         |
|                                                                             |
|o37 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i38 : J2 = I1 + J                                                            |
|                                                                             |
|                                                                             |
|o38 = ideal (727710p          - 2870478p          + 1936669p         ,       |
|                    {1, 0, 1}           {1, 1, 0}           {1, 1, 1}        |
|      -----------------------------------------------------------------------|
|                                                                             |
|      1188593p          + 9157515p          - 12232731p         ,            |
|              {1, 0, 0}           {1, 1, 0}            {1, 1, 1}             |
|      -----------------------------------------------------------------------|
|                                                                             |
|      437292065p          - 70326711p          - 91417042p         ,         |
|                {0, 1, 1}            {1, 1, 0}            {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|                                                                             |
|      24988118p          - 85869p          - 10485198p         ,             |
|               {0, 1, 0}         {1, 1, 0}            {1, 1, 1}              |
|      -----------------------------------------------------------------------|
|                                                                             |
|      22730129187225p          + 9420433731891p          - 24975873351298p   |
|                     {0, 0, 1}                 {1, 1, 0}                  {1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           , 21214787241410p          - 34552109032893p          +           |
|      1, 1}                 {0, 0, 0}                  {1, 1, 0}             |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      22940456357154p         , 140653422p          + 179828669p         p   |
|                     {1, 1, 1}            {1, 1, 0}             {1, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|                        2                                                    |
|            - 366981930p         , p          + p          + p          +    |
|      1, 1}             {1, 1, 1}   {0, 0, 0}    {0, 0, 1}    {0, 1, 0}      |
|      -----------------------------------------------------------------------|
|      p          + p          + p          + p          + p          - 1)    |
|       {0, 1, 1}    {1, 0, 0}    {1, 0, 1}    {1, 1, 0}    {1, 1, 1}         |
|                                                                             |
|o38 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+

Acknowledgement
===============

We thank Ben Hollering<https://sites.google.com/view/benhollering> and Mahrud
Sayrafi<https://www-users.cse.umn.edu/~mahrud/> for their support during the
Macaulay2 in the Sciences
Workshop<https://www.mis.mpg.de/de/events/series/macaulay2-in-the-sciences>
where the development of this package began.

Contributors
============

The following people have generously contributed their time and effort to this
project: Luca Sodomaco<https://sites.google.com/view/luca-sodomaco/home>.

References
==========

This package is based on the following papers:

- Nash Equilibria: [H. Abo, I. Portakal, and L. Sodomaco: A vector bundle
approach to Nash equilibria (see https://arxiv.org/abs/2504.03456 )] available
on arXiv.

- Correlated Equilibria: [M.-C. Brandenburg, B. Hollering, I. Portakal:
Combinatorics of Correlated Equilibria (see
https://www.tandfonline.com/doi/full/10.1080/10586458.2024.2340000 )]
Experimental Mathematics, 2024.

- Dependency Equilibria: [I. Portakal and B. Sturmfels: Geometry of dependency
equilibria (see https://www.openstarts.units.it/server/api/core/bitstreams/
12c8c6f4-d535-49e3-b6a5-d4f12c331b0f/content )] published in Rend. Istit. Mat.
Univ. Trieste 54 (Art. No. 5), 2022, 13, 2022.

- Conditional Independence Equilibria: [I. Portakal and J. Sendra-Arranz: Game
theory of undirected graphical models (see
https://www.sciencedirect.com/science/article/pii/S0021869324006707 )] Journal
of Algebra, Volume 666, 2025.

Caveat
======

GameTheory uses Polyhedra.m2 for the methods of correlated equilibria and
GraphicalModels.m2 for the methods of conditional independence equilibria.
Throughout the package, we followed Macaulay2's convention of zero-based
indexing. This can be seen e.g., in the methods of nashEquilibriumRing,
and probabilityRing. In
particular, for an $n$-player game, the players are labeled $0, \ldots, n-1$,
and if player $i$ has $d_i$ pure strategies, they are labeled $0, \ldots,
d_i-1$.

Authors
=======

  * Erin Connelly (see https://erinconnelly96.github.io/ )
    <erin.connelly@uni-osnabrueck.de>
  * Vincenzo Galgano (see https://sites.google.com/view/vincenzogalgano )
    <galgano@mpi-cbg.de>
  * Zhuang He (see https://sites.google.com/view/hezhuang/home )
    <zhuang.he@unito.it>
  * Lars Kastner (see https://lkastner.github.io ) <kastner@math.tu-berlin.de>
  * Giacomo Maletto (see https://www.kth.se/profile/gmaletto )
    <gmaletto@kth.se>
  * Elke Neuhaus (see https://sites.google.com/view/elkeneuhaus )
    <elke@neuhaus-rp.de>
  * Irem Portakal (see https://www.irem-portakal.de ) <mail@irem-portakal.de>
  * Hannah Tillmann-Morris (see
    https://sites.google.com/view/hannah-tillmann-morris )
    <tillmann@mis.mpg.de>
  * Chenyang Zhao (see https://www.felixzhao.com/ ) <cz2922@ic.ac.uk>

Version
=======

This documentation describes version 1.0 of GameTheory.

Citation
========

If you have used this package in your research, please cite it as follows:

+-----------------------------------------------------------------------------+
|@misc{GameTheorySource,                                                      |
|  title = {{GameTheory: computing equilibria in game theory. Version~1.0}},  |
|  author = {Erin Connelly and Vincenzo Galgano and Zhuang He and Lars Kastner|
|  howpublished = {A \emph{Macaulay2} package available at                    |
|    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}} |
|}                                                                            |
|-----------------------------------------------------------------------------|
|and Giacomo Maletto and Elke Neuhaus and Irem Portakal and Hannah            |
|-----------------------------------------------------------------------------|
|Tillmann-Morris and Chenyang Zhao},                                          |
+-----------------------------------------------------------------------------+

Exports
=======

  * Types
      * Tensor -- a mutable hash table representing a tensor
  * Functions and commands
      * assemblePlayeriPolynomials -- get
        all incentive constraint polynomials for a player
      * assemblePolynomial -- compute incentive
        constraint polynomial for deviation
      * "blockDerangements" -- see blockDerangements(List),
        -- compute the block derangements
      * ciIdeal -- the ideal of a list of conditional
        independence statements
      * correlatedEquilibria -- compute the
        correlated equilibrium polytope for a game
      * "deltaList" -- see deltaList(List), --
        generate the list of Newton polytopes for a generic game
      * enumerateTensorIndices -- generate index
        tuples for a tensor with given dimensions
      * getVariableToIndexset -- retrieve a
        polynomial ring variable by its index tuple
      * indexset -- get the list of index tuples of a tensor
      * intersectWithCImodel -- ideal of the
        intersection of a given variety with the conditional independence model
      * konstanzMatrix -- construct the Konstanz matrix
        of a given game
      * mapToMarkovRing -- ring isomorphism from the
        given probabilityRing to the corresponding markovRing
      * mapToProbabilityRing -- ring isomorphism
        to the given probabilityRing from the corresponding markovRing
      * "nashEquilibriumIdeal" -- see nashEquilibriumIdeal(Ring,List),
        -- make the Nash Equilibrium
        ideal
      * "nashEquilibriumRing" -- see nashEquilibriumRing(List),
        -- define the Nash Equilibrium ring
      * "numberTMNE" -- see numberTMNE(List), --
        compute the maximum number of totally mixed Nash equilibria for a
        generic game
      * probabilityRing -- ring of probability
        distributions of a game indexed by ordered multi-indices
      * probabilitySumIdeal -- ideal enforcing that
        a probability distribution sums to 1
      * randomGame -- construct a game of a given format
        with arbitrary payoffs
      * randomTensor -- construct a tensor with random
        entries from a given ring
      * slice -- extract a slice of a tensor
      * spohnCI -- ideal of the Spohn conditional independence
        (CI) variety
      * spohnIdeal -- compute the ideal of the Spohn variety
        of a given game
      * spohnMatrices -- compute the list of Spohn
        matrices of a given game
      * toMarkovRing -- ring of joint probability
        distributions created with the markovRing function from the
        GraphicalModels package
      * zeroTensor -- construct a tensor with zero entries
        from a given ring.
  * Methods
      * "assemblePlayeriPolynomials(Ring,Tensor,ZZ)" -- see
        assemblePlayeriPolynomials, -- get all
        incentive constraint polynomials for a player
      * "assemblePolynomial(Ring,Tensor,List)" -- see assemblePolynomial,
        -- compute incentive constraint polynomial for
        deviation
      * blockDerangements(List) -- compute
        the block derangements
      * "ciIdeal(Ring,Graph)" -- see ciIdeal, -- the ideal of a
        list of conditional independence statements
      * "ciIdeal(Ring,Graph,List)" -- see ciIdeal, -- the ideal
        of a list of conditional independence statements
      * "ciIdeal(Ring,List)" -- see ciIdeal, -- the ideal of a
        list of conditional independence statements
      * "ciIdeal(Ring,List,List)" -- see ciIdeal, -- the ideal
        of a list of conditional independence statements
      * coefficientRing(Tensor) (missing documentation)
      * "correlatedEquilibria(List)" -- see correlatedEquilibria,
        -- compute the correlated equilibrium polytope
        for a game
      * deltaList(List) -- generate the list of
        Newton polytopes for a generic game
      * "enumerateTensorIndices(List)" -- see enumerateTensorIndices,
        -- generate index tuples for a tensor with
        given dimensions
      * "enumerateTensorIndices(ZZ)" -- see enumerateTensorIndices,
        -- generate index tuples for a tensor with
        given dimensions
      * format(Tensor) (missing documentation)
      * "getVariableToIndexset(Ring,List)" -- see getVariableToIndexset,
        -- retrieve a polynomial ring variable by its
        index tuple
      * "indexset(Tensor)" -- see indexset, -- get the list of
        index tuples of a tensor
      * "intersectWithCImodel(Ideal,Graph)" -- see intersectWithCImodel,
        -- ideal of the intersection of a given variety
        with the conditional independence model
      * "intersectWithCImodel(Ideal,Graph,List)" -- see
        intersectWithCImodel, -- ideal of the
        intersection of a given variety with the conditional independence model
      * "intersectWithCImodel(Ideal,List)" -- see intersectWithCImodel,
        -- ideal of the intersection of a given variety
        with the conditional independence model
      * "intersectWithCImodel(Ideal,List,List)" -- see
        intersectWithCImodel, -- ideal of the
        intersection of a given variety with the conditional independence model
      * konstanzMatrix(Ring,List) (missing documentation)
      * "mapToMarkovRing(Ring)" -- see mapToMarkovRing,
        -- ring isomorphism from the given probabilityRing to the corresponding
        markovRing
      * "mapToProbabilityRing(Ring)" -- see mapToProbabilityRing,
        -- ring isomorphism to the given probabilityRing
        from the corresponding markovRing
      * nashEquilibriumIdeal(Ring,List)
        -- make the Nash Equilibrium
        ideal
      * nashEquilibriumRing(List), --
        define the Nash Equilibrium ring
      * numberTMNE(List) -- compute the maximum
        number of totally mixed Nash equilibria for a generic game
      * "probabilityRing(List)" -- see probabilityRing,
        -- ring of probability distributions of a game indexed by ordered
        multi-indices
      * "probabilitySumIdeal(Ring)" -- see probabilitySumIdeal,
        -- ideal enforcing that a probability distribution
        sums to 1
      * randomGame(List) (missing documentation)
      * "randomTensor(List)" -- see randomTensor, --
        construct a tensor with random entries from a given ring
      * "randomTensor(Ring,List)" -- see randomTensor, --
        construct a tensor with random entries from a given ring
      * "slice(Tensor,List,List)" -- see slice, -- extract a slice
        of a tensor
      * "spohnCI(Ring,List,Graph)" -- see spohnCI, -- ideal of
        the Spohn conditional independence (CI) variety
      * "spohnCI(Ring,List,Graph,List)" -- see spohnCI, -- ideal
        of the Spohn conditional independence (CI) variety
      * "spohnCI(Ring,List,List)" -- see spohnCI, -- ideal of
        the Spohn conditional independence (CI) variety
      * "spohnCI(Ring,List,List,List)" -- see spohnCI, -- ideal
        of the Spohn conditional independence (CI) variety
      * spohnIdeal(Ring,List) (missing documentation)
      * spohnMatrices(Ring,List) (missing documentation)
      * "toMarkovRing(Ring)" -- see toMarkovRing, -- ring
        of joint probability distributions created with the markovRing function
        from the GraphicalModels package
      * "zeroTensor(List)" -- see zeroTensor, -- construct a
        tensor with zero entries from a given ring.
      * "zeroTensor(Ring,List)" -- see zeroTensor, --
        construct a tensor with zero entries from a given ring.
  * Symbols
      * KonstanzVariableName (missing documentation)
      * ProbabilityVariableName (missing documentation)

For the programmer
==================

The object GameTheory, is a package (Macaulay2Doc),
defined in GameTheory.m2.

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:914:0.

assemblePlayeriPolynomials -- get all incentive constraint polynomials for a player
***********************************************************************************

  * Usage:
        assemblePlayeriPolynomials(PR, Xi, i)
  * Inputs:
      * PR, a ring (Macaulay2Doc), Polynomial ring of
        probabilities.
      * Xi, an instance of the type Tensor, Tensor of strategy
        probabilities.
      * i, an integer (Macaulay2Doc), The player index.
  * Outputs:
      * a list (Macaulay2Doc), A list of polynomials corresponding
        to the player's deviation constraints.

Description
===========

For a given player $i$, this method computes all incentive constraint
polynomials $\forall k,l \in [d_i]$, representing deviations from strategy $k$
to $l$.

+---------------------------------------------------------+
|i1 : R = QQ[p_{0,0}, p_{0,1}, p_{1,0}, p_{1,1}];         |
+---------------------------------------------------------+
|i2 : Xi = randomTensor(QQ, {2,2});                       |
+---------------------------------------------------------+
|i3 : assemblePlayeriPolynomials(R, Xi, 0)                |
|                                                         |
|         15          1           15          1           |
|o3 = {0, --p       + -p      , - --p       - -p      , 0}|
|          4 {0, 0}   2 {0, 1}     4 {1, 0}   2 {1, 1}    |
|                                                         |
|o3 : List                                                |
+---------------------------------------------------------+

See also
========

  * probabilityRing -- ring of probability
    distributions of a game indexed by ordered multi-indices
  * assemblePolynomial -- compute incentive
    constraint polynomial for deviation
  * correlatedEquilibria -- compute the correlated
    equilibrium polytope for a game

Ways to use assemblePlayeriPolynomials:
=======================================

  * "assemblePlayeriPolynomials(Ring,Tensor,ZZ)"

For the programmer
==================

The object assemblePlayeriPolynomials, is a
method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1254:0.

assemblePolynomial -- compute incentive constraint polynomial for deviation
***************************************************************************

  * Usage:
        assemblePolynomial(PR, Xi, {i,k,l})
  * Inputs:
      * PR, a ring (Macaulay2Doc), Polynomial ring of strategy
        probabilities.
      * Xi, an instance of the type Tensor, Tensor of strategy
        probabilities.
      * {i, k, l}, a list (Macaulay2Doc), A triple where: - $i$ is
        the player index, - $k$ is the current strategy, - $l$ is the deviating
        strategy.
  * Outputs:
      * a ring element (Macaulay2Doc), A polynomial
        representing the incentive constraint for player $i$ deviating from $k$
        to $l$.

Description
===========

This function computes a linear inequality encoding the condition for a
correlated equilibrium: the expected utility from playing $k$ should be no less
than from playing $l$.

+------------------------------------------------+
|i1 : R = QQ[p_{0,0}, p_{0,1}, p_{1,0}, p_{1,1}];|
+------------------------------------------------+
|i2 : Xi = randomTensor(QQ, {2,2});              |
+------------------------------------------------+
|i3 : assemblePolynomial(R, Xi, {0,0,1})         |
|                                                |
|     15          1                              |
|o3 = --p       + -p                             |
|      4 {0, 0}   2 {0, 1}                       |
|                                                |
|o3 : R                                          |
+------------------------------------------------+

See also
========

  * probabilityRing -- ring of probability
    distributions of a game indexed by ordered multi-indices
  * getVariableToIndexset -- retrieve a
    polynomial ring variable by its index tuple
  * assemblePlayeriPolynomials -- get all
    incentive constraint polynomials for a player

Ways to use assemblePolynomial:
===============================

  * "assemblePolynomial(Ring,Tensor,List)"

For the programmer
==================

The object assemblePolynomial, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1217:0.

blockDerangements(List) -- compute the block derangements
*********************************************************

  * Function: blockDerangements,
  * Usage:
        blockDerangements D
  * Inputs:
      * D, a list (Macaulay2Doc), a list of integers representing
        the format of the game
  * Outputs:
      * a list (Macaulay2Doc), a list of block derangements with
        respect to D.

Description
===========

Given a partition $\{F_0,\ldots,F_{n-1}\}$ of a finite set $F$, a block
derangement of $F$ with respect to $\{F_0,\ldots,F_{n-1}\}$ is a permutation
$P\colon F\to F$ of $F$ such that $P(F_i)\cap F_i=\emptyset$ for every
$i\in\{0,\ldots,n-1\}$.

The function considers the input $D=(d_0,\ldots,d_{n-1})$, defines the set
$F=F_0\cup\cdots\cup F_{n-1}$, where $F_i=\{(i,j)\mid j\in\{0,\ldots,d_i-2\}\}$
for every $i\in\{0,\ldots,n-1\}$, and computes the set of permutations of F.
Then, it creates an empty list BD, and for each permutation $P$, if $P(F_i)\cap
F_i=\emptyset$ for every $i\in\{0,\ldots,n-1\}$, then $P$ is added to the list
BD. The function returns the list BD.

The number of elements of BD corresponds to the maximum number of totally mixed
Nash equilibria of a generic game of a given format.

+---------------------------------------------------------------------+
|i1 : D = {3,3,3};                                                    |
+---------------------------------------------------------------------+
|i2 : BD = blockDerangements D;                                       |
+---------------------------------------------------------------------+
|i3 : netList BD                                                      |
|                                                                     |
|     +--------------------+--------------------+--------------------+|
|o3 = |set {(2, 0), (1, 1)}|set {(2, 1), (0, 1)}|set {(0, 0), (1, 0)}||
|     +--------------------+--------------------+--------------------+|
|     |set {(2, 1), (2, 0)}|set {(0, 0), (0, 1)}|set {(1, 0), (1, 1)}||
|     +--------------------+--------------------+--------------------+|
|     |set {(1, 0), (2, 0)}|set {(2, 1), (0, 0)}|set {(0, 1), (1, 1)}||
|     +--------------------+--------------------+--------------------+|
|     |set {(2, 1), (1, 1)}|set {(0, 0), (2, 0)}|set {(1, 0), (0, 1)}||
|     +--------------------+--------------------+--------------------+|
|     |set {(1, 0), (2, 0)}|set {(2, 1), (0, 1)}|set {(0, 0), (1, 1)}||
|     +--------------------+--------------------+--------------------+|
|     |set {(2, 1), (1, 1)}|set {(0, 1), (2, 0)}|set {(0, 0), (1, 0)}||
|     +--------------------+--------------------+--------------------+|
|     |set {(1, 0), (1, 1)}|set {(2, 1), (2, 0)}|set {(0, 0), (0, 1)}||
|     +--------------------+--------------------+--------------------+|
|     |set {(2, 1), (1, 0)}|set {(0, 0), (2, 0)}|set {(0, 1), (1, 1)}||
|     +--------------------+--------------------+--------------------+|
|     |set {(2, 0), (1, 1)}|set {(2, 1), (0, 0)}|set {(1, 0), (0, 1)}||
|     +--------------------+--------------------+--------------------+|
|     |set {(2, 1), (1, 0)}|set {(0, 1), (2, 0)}|set {(0, 0), (1, 1)}||
|     +--------------------+--------------------+--------------------+|
+---------------------------------------------------------------------+

See also
========

  * deltaList -- generate the list of Newton
    polytopes for a generic game
  * numberTMNE -- compute the maximum number of
    totally mixed Nash equilibria for a generic game

Ways to use this method:
========================

  * blockDerangements(List) -- compute the
    block derangements
-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1494:0.

ciIdeal -- the ideal of a list of conditional independence statements
*********************************************************************

  * Usage:
        ciIdeal (R, Stmts)
        ciIdeal (R, G)
        ciIdeal (R, Stmts, PlayerNames)
        ciIdeal (R, G, PlayerNames)
  * Inputs:
      * R, a ring (Macaulay2Doc), must be created using
        probabilityRing
      * Stmts, a list (Macaulay2Doc), the list of conditional
        independence statements
      * G, a graph (Graphs), the graph modelling the conditional
        dependencies between players
      * PlayerNames, a list (Macaulay2Doc), the ordered list of
        players - the names of the random variables in the conditional
        independence statements or vertices of the graph. If PlayerNames is
        omitted, the players (or the vertices of G) are assumed to be labelled
        1..n.
  * Outputs:
      * an ideal (Macaulay2Doc), the ideal in R of conditional
        independence relations

Description
===========

ciIdeal computes the ideal of a list of conditional independence statements.
The input can be the list of conditional independence statements itself, or a
graph modelling the conditional dependencies between players. This method is
the same as conditionalIndependenceIdeal (GraphicalModels),
from GameTheory.m2, included
here for convenience and compatibility with this package.

A single conditional independence statement is a list consisting of three
disjoint lists of indices for random variables, e.g. $\{ \{1,2\},\{4\}, \{3\}
\}$ which represents the conditional independence statement ``$(X_1, X_2)$ is
conditionally independent of $X_4$ given $X_3$''. Given an undirected graph
$G$, the conditional independence statements are produced via the globalMarkov
function from the GraphicalModels package. A global Markov statement for $G$ is
a list $\{A, B, C\}$ of three disjoint lists of vertices of $G$, where the
subset $C$ separates the subset $A$ from the subset $B$ in the graph $G$.

The output is an ideal of the given ring PR, which must be created using the
probabilityRing function. This function computes the ideal using the
conditionalIndependenceIdeal (GraphicalModels),
function from the GraphicalModels package, then maps it to an ideal of PR via
the mapToProbabilityRing function.

+-----------------------------------------------------------------------------+
|i1 : FF = ZZ/32003                                                           |
|                                                                             |
|o1 = FF                                                                      |
|                                                                             |
|o1 : QuotientRing                                                            |
+-----------------------------------------------------------------------------+
|i2 : d = {2,3,2};                                                            |
+-----------------------------------------------------------------------------+
|i3 : PR = probabilityRing (d, CoefficientRing => FF);                        |
+-----------------------------------------------------------------------------+
|i4 : G = graph ({}, Singletons => {1,2,3});                                  |
+-----------------------------------------------------------------------------+
|i5 : I = ciIdeal (PR, G)                                                     |
|                                                                             |
|o5 = ideal (- p         p          + p         p         , - p         p     |
|               {0, 0, 1} {1, 0, 0}    {0, 0, 0} {1, 0, 1}     {0, 1, 0} {1,  |
|     ------------------------------------------------------------------------|
|           + p         p         , - p         p          + p         p      |
|     0, 0}    {0, 0, 0} {1, 1, 0}     {0, 1, 0} {1, 0, 1}    {0, 0, 1} {1, 1,|
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     0}     {0, 1, 1} {1, 0, 0}    {0, 0, 0} {1, 1, 1}     {0, 1, 1} {1, 0,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     1}    {0, 0, 1} {1, 1, 1}     {0, 1, 1} {1, 1, 0}    {0, 1, 0} {1, 1,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     1}     {0, 2, 0} {1, 0, 0}    {0, 0, 0} {1, 2, 0}     {0, 2, 0} {1, 0,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     1}    {0, 0, 1} {1, 2, 0}     {0, 2, 0} {1, 1, 0}    {0, 1, 0} {1, 2,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     0}     {0, 2, 0} {1, 1, 1}    {0, 1, 1} {1, 2, 0}     {0, 2, 1} {1, 0,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     0}    {0, 0, 0} {1, 2, 1}     {0, 2, 1} {1, 0, 1}    {0, 0, 1} {1, 2,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     1}     {0, 2, 1} {1, 1, 0}    {0, 1, 0} {1, 2, 1}     {0, 2, 1} {1, 1,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     1}    {0, 1, 1} {1, 2, 1}     {0, 2, 1} {1, 2, 0}    {0, 2, 0} {1, 2,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     1}     {0, 0, 1} {0, 1, 0}    {0, 0, 0} {0, 1, 1}     {0, 1, 0} {1, 0,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     0}    {0, 0, 0} {1, 1, 0}     {0, 1, 1} {1, 0, 0}    {0, 0, 1} {1, 1,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     0}     {0, 1, 0} {1, 0, 1}    {0, 0, 0} {1, 1, 1}     {0, 1, 1} {1, 0,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     1}    {0, 0, 1} {1, 1, 1}     {1, 0, 1} {1, 1, 0}    {1, 0, 0} {1, 1,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     1}     {0, 0, 1} {0, 2, 0}    {0, 0, 0} {0, 2, 1}     {0, 2, 0} {1, 0,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     0}    {0, 0, 0} {1, 2, 0}     {0, 2, 1} {1, 0, 0}    {0, 0, 1} {1, 2,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     0}     {0, 2, 0} {1, 0, 1}    {0, 0, 0} {1, 2, 1}     {0, 2, 1} {1, 0,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     1}    {0, 0, 1} {1, 2, 1}     {1, 0, 1} {1, 2, 0}    {1, 0, 0} {1, 2,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     1}     {0, 1, 1} {0, 2, 0}    {0, 1, 0} {0, 2, 1}     {0, 2, 0} {1, 1,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     0}    {0, 1, 0} {1, 2, 0}     {0, 2, 1} {1, 1, 0}    {0, 1, 1} {1, 2,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     0}     {0, 2, 0} {1, 1, 1}    {0, 1, 0} {1, 2, 1}     {0, 2, 1} {1, 1,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     1}    {0, 1, 1} {1, 2, 1}     {1, 1, 1} {1, 2, 0}    {1, 1, 0} {1, 2,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     1}     {0, 0, 1} {0, 1, 0}    {0, 0, 0} {0, 1, 1}     {0, 0, 1} {0, 2,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     0}    {0, 0, 0} {0, 2, 1}     {0, 1, 1} {0, 2, 0}    {0, 1, 0} {0, 2,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     1}     {0, 0, 1} {1, 0, 0}    {0, 0, 0} {1, 0, 1}     {0, 1, 1} {1, 0,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     0}    {0, 1, 0} {1, 0, 1}     {0, 2, 1} {1, 0, 0}    {0, 2, 0} {1, 0,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     1}     {0, 0, 1} {1, 1, 0}    {0, 0, 0} {1, 1, 1}     {0, 1, 1} {1, 1,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     0}    {0, 1, 0} {1, 1, 1}     {0, 2, 1} {1, 1, 0}    {0, 2, 0} {1, 1,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     1}     {1, 0, 1} {1, 1, 0}    {1, 0, 0} {1, 1, 1}     {0, 0, 1} {1, 2,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     0}    {0, 0, 0} {1, 2, 1}     {0, 1, 1} {1, 2, 0}    {0, 1, 0} {1, 2,   |
|     ------------------------------------------------------------------------|
|       , - p         p          + p         p         , - p         p        |
|     1}     {0, 2, 1} {1, 2, 0}    {0, 2, 0} {1, 2, 1}     {1, 0, 1} {1, 2,  |
|     ------------------------------------------------------------------------|
|        + p         p         , - p         p          + p         p         |
|     0}    {1, 0, 0} {1, 2, 1}     {1, 1, 1} {1, 2, 0}    {1, 1, 0} {1, 2,   |
|     ------------------------------------------------------------------------|
|       )                                                                     |
|     1}                                                                      |
|                                                                             |
|o5 : Ideal of PR                                                             |
+-----------------------------------------------------------------------------+

Here is an example where the vertices of the graph need to be relabeled.

+-----------------------------------------------------------------------------+
|i6 : FF = ZZ/32003                                                           |
|                                                                             |
|o6 = FF                                                                      |
|                                                                             |
|o6 : QuotientRing                                                            |
+-----------------------------------------------------------------------------+
|i7 : d = {2,3,2};                                                            |
+-----------------------------------------------------------------------------+
|i8 : PR = probabilityRing (d, CoefficientRing => FF);                        |
+-----------------------------------------------------------------------------+
|i9 : G = graph {{John,Matthew},{Matthew,Sarah}};                             |
+-----------------------------------------------------------------------------+
|i10 : I = ciIdeal (PR, G, {John,Matthew,Sarah})                              |
|                                                                             |
|o10 = ideal (- p         p          + p         p         , - p         p    |
|                {0, 0, 1} {1, 0, 0}    {0, 0, 0} {1, 0, 1}     {0, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|            + p         p         , - p         p          + p         p     |
|      1, 0}    {0, 1, 0} {1, 1, 1}     {0, 2, 1} {1, 2, 0}    {0, 2, 0} {1,  |
|      -----------------------------------------------------------------------|
|           )                                                                 |
|      2, 1}                                                                  |
|                                                                             |
|o10 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+

Here is an example where the conditional independence relations are given with
a List.

+-----------------------------------------------------------------------------+
|i11 : FF = ZZ/32003                                                          |
|                                                                             |
|o11 = FF                                                                     |
|                                                                             |
|o11 : QuotientRing                                                           |
+-----------------------------------------------------------------------------+
|i12 : d = {2,3,2};                                                           |
+-----------------------------------------------------------------------------+
|i13 : PR = probabilityRing (d, CoefficientRing => FF);                       |
+-----------------------------------------------------------------------------+
|i14 : G = graph {{1,2},{2,3}};                                               |
+-----------------------------------------------------------------------------+
|i15 : L = {{{1},{3},{2}}}                                                    |
|                                                                             |
|o15 = {{{1}, {3}, {2}}}                                                      |
|                                                                             |
|o15 : List                                                                   |
+-----------------------------------------------------------------------------+
|i16 : I1 = ciIdeal (PR,G)                                                    |
|                                                                             |
|o16 = ideal (- p         p          + p         p         , - p         p    |
|                {0, 0, 1} {1, 0, 0}    {0, 0, 0} {1, 0, 1}     {0, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|            + p         p         , - p         p          + p         p     |
|      1, 0}    {0, 1, 0} {1, 1, 1}     {0, 2, 1} {1, 2, 0}    {0, 2, 0} {1,  |
|      -----------------------------------------------------------------------|
|           )                                                                 |
|      2, 1}                                                                  |
|                                                                             |
|o16 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i17 : I2 = ciIdeal (PR,L)                                                    |
|                                                                             |
|o17 = ideal (- p         p          + p         p         , - p         p    |
|                {0, 0, 1} {1, 0, 0}    {0, 0, 0} {1, 0, 1}     {0, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|            + p         p         , - p         p          + p         p     |
|      1, 0}    {0, 1, 0} {1, 1, 1}     {0, 2, 1} {1, 2, 0}    {0, 2, 0} {1,  |
|      -----------------------------------------------------------------------|
|           )                                                                 |
|      2, 1}                                                                  |
|                                                                             |
|o17 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i18 : I1 == I2                                                               |
|                                                                             |
|o18 = true                                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * conditionalIndependenceIdeal (GraphicalModels)
    -- ideal of a list of
    conditional independent statements
  * mapToProbabilityRing -- ring isomorphism to
    the given probabilityRing from the corresponding markovRing
  * toMarkovRing -- ring of joint probability
    distributions created with the markovRing function from the GraphicalModels
    package
  * ciIdeal -- the ideal of a list of conditional independence
    statements
  * globalMarkov (GraphicalModels) -- global Markov
    statements for a graph or a directed graph

Ways to use ciIdeal:
====================

  * "ciIdeal(Ring,Graph)"
  * "ciIdeal(Ring,Graph,List)"
  * "ciIdeal(Ring,List)"
  * "ciIdeal(Ring,List,List)"

For the programmer
==================

The object ciIdeal, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1947:0.

correlatedEquilibria -- compute the correlated equilibrium polytope for a game
******************************************************************************

  * Usage:
        correlatedEquilibria X
  * Inputs:
      * X, a list (Macaulay2Doc), A list of payoff tensors, one for
        each player.
  * Outputs:
      * a convex polyhedron (Polyhedra), The polytope
        representing the set of correlated equilibria for the game.

Description
===========

This method constructs and returns the correlated equilibrium polytope for a
finite game. The input is a list of payoff tensors, one for each player. The
tensor at position i gives the payoffs for player i.

+--------------------------------------------------------------+
|i1 : X1 = zeroTensor(QQ, {2,2});                              |
+--------------------------------------------------------------+
|i2 : X2 = zeroTensor(QQ, {2,2});                              |
+--------------------------------------------------------------+
|i3 : X1#{0,0} = -99; X1#{0,1} = 1; X1#{1,0} = 0; X1#{1,1} = 0;|
+--------------------------------------------------------------+
|i7 : X2#{0,0} = -99; X2#{0,1} = 0; X2#{1,0} = 1; X2#{1,1} = 0;|
+--------------------------------------------------------------+
|i11 : CE = correlatedEquilibria {X1, X2}                      |
|                                                              |
|o11 = CE                                                      |
|                                                              |
|o11 : Polyhedron                                              |
+--------------------------------------------------------------+
|i12 : vertices CE                                             |
|                                                              |
|o12 = | 0 0 1/199  0      1/10000    |                        |
|      | 1 0 99/199 1/101  99/10000   |                        |
|      | 0 1 99/199 1/101  99/10000   |                        |
|      | 0 0 0      99/101 9801/10000 |                        |
|                                                              |
|               4       5                                      |
|o12 : Matrix QQ  <-- QQ                                       |
+--------------------------------------------------------------+
|i13 : facets CE                                               |
|                                                              |
|o13 = (| -1 0   0   0  |, 0)                                  |
|       | 99 -1  0   0  |                                      |
|       | 99 0   -1  0  |                                      |
|       | 0  0   0   -1 |                                      |
|       | 0  -99 0   1  |                                      |
|       | 0  0   -99 1  |                                      |
|                                                              |
|o13 : Sequence                                                |
+--------------------------------------------------------------+
|i14 : fVector CE                                              |
|                                                              |
|o14 = {5, 9, 6, 1}                                            |
|                                                              |
|o14 : List                                                    |
+--------------------------------------------------------------+
+----------------------------------+
|i15 : X = randomGame {2,2,3};     |
+----------------------------------+
|i16 : CE = correlatedEquilibria X;|
+----------------------------------+
|i17 : dim CE                      |
|                                  |
|o17 = 11                          |
+----------------------------------+

See also
========

  * assemblePolynomial -- compute incentive
    constraint polynomial for deviation
  * assemblePlayeriPolynomials -- get all
    incentive constraint polynomials for a player

Ways to use correlatedEquilibria:
=================================

  * "correlatedEquilibria(List)"

For the programmer
==================

The object correlatedEquilibria, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1297:0.

deltaList(List) -- generate the list of Newton polytopes for a generic game
***************************************************************************

  * Function: deltaList,
  * Usage:
        deltaList d
  * Inputs:
      * d, a list (Macaulay2Doc), a list of positive integers,
        representing the format of the game
  * Outputs:
      * a list (Macaulay2Doc), a list of polytopes formed by taking
        the product of certain convex sets and simplices

Description
===========

For a generic $n$-player game where the $i$-th player has $d_i$ pure
strategies, the maximum number of isolated totally mixed Nash equilibria is
given by the mixed volume of the following list of polytopes:

\[ (\Delta^{(1)}, \cdots, \Delta^{(1)},\Delta^{(2)}, \cdots, \Delta^{(2)},
\cdots, \Delta^{(n)}, \cdots, \Delta^{(n)}),\]

where each $\Delta^{(i)}$ repeats itself $d_i - 1$ times, and is the product of
simplices

\[ \Delta^{(i)} := \Delta_{d_{1}-1}\times \Delta_{d_{2}-1} \times \cdots \times
\Delta_{d_{i-1}-1} \times \{0\} \times \Delta_{d_{i+1}-1} \times \cdots \times
\Delta_{d_{n}-1}.\]

This function constructs and returns this list of polytopes. Each
$\Delta^{(i)}$ is a polytope in an ambient vector space of dimension
$d_1+d_2+\cdots+d_{n}-n$.

+--------------------------------------------------------------------+
|i1 : DL = deltaList {2,2,2}                                         |
|                                                                    |
|o1 = {Polyhedron{...1...}, Polyhedron{...1...}, Polyhedron{...1...}}|
|                                                                    |
|o1 : List                                                           |
+--------------------------------------------------------------------+

Each entries of DL is a polytope of dimension 2, in a $2+2+2-3=3$ dimensional
vector space.

+-----------------------------+
|i2 : apply(DL, p -> dim p)   |
|                             |
|o2 = {2, 2, 2}               |
|                             |
|o2 : List                    |
+-----------------------------+
|i3 : apply(DL, p -> ambDim p)|
|                             |
|o3 = {3, 3, 3}               |
|                             |
|o3 : List                    |
+-----------------------------+

See also
========

  * nashEquilibriumRing -- define the
    Nash Equilibrium ring
  * nashEquilibriumIdeal -- make
    the Nash Equilibrium ideal

Ways to use this method:
========================

  * deltaList(List) -- generate the list of Newton
    polytopes for a generic game
-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1413:0.

enumerateTensorIndices -- generate index tuples for a tensor with given dimensions
**********************************************************************************

  * Usage:
        enumerateTensorIndices d
  * Inputs:
      * d, an integer (Macaulay2Doc), A single integer.
      * d, a list (Macaulay2Doc), A list of integers, representing
        the format of a tensor.
  * Outputs:
      * a list (Macaulay2Doc), A list of index tuples, where each
        tuple corresponds to a coordinate of the tensor.

Description
===========

This function generates all possible index tuples for a tensor with the
specified dimensions. For a tensor of format $d_1 \times d_2 \cdots d_n$, it
returns a list of all tuples $(i_1, i2, \ldots, i_n)$ where $0 \leq i_j < d_j$.

+-----------------------------------------------------------------------------+
|i1 : enumerateTensorIndices 5                                                |
|                                                                             |
|o1 = {{0}, {1}, {2}, {3}, {4}}                                               |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+
|i2 : enumerateTensorIndices {2,2}                                            |
|                                                                             |
|o2 = {{0, 0}, {0, 1}, {1, 0}, {1, 1}}                                        |
|                                                                             |
|o2 : List                                                                    |
+-----------------------------------------------------------------------------+
|i3 : enumerateTensorIndices {3,2,4}                                          |
|                                                                             |
|o3 = {{0, 0, 0}, {0, 0, 1}, {0, 0, 2}, {0, 0, 3}, {0, 1, 0}, {0, 1, 1}, {0,  |
|     ------------------------------------------------------------------------|
|     1, 2}, {0, 1, 3}, {1, 0, 0}, {1, 0, 1}, {1, 0, 2}, {1, 0, 3}, {1, 1, 0},|
|     ------------------------------------------------------------------------|
|     {1, 1, 1}, {1, 1, 2}, {1, 1, 3}, {2, 0, 0}, {2, 0, 1}, {2, 0, 2}, {2, 0,|
|     ------------------------------------------------------------------------|
|     3}, {2, 1, 0}, {2, 1, 1}, {2, 1, 2}, {2, 1, 3}}                         |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * Tensor -- a mutable hash table representing a tensor
  * zeroTensor -- construct a tensor with zero entries from
    a given ring.
  * randomTensor -- construct a tensor with random entries
    from a given ring

Ways to use enumerateTensorIndices:
===================================

  * "enumerateTensorIndices(List)"
  * "enumerateTensorIndices(ZZ)"

For the programmer
==================

The object enumerateTensorIndices, is a
method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:953:0.

getVariableToIndexset -- retrieve a polynomial ring variable by its index tuple
*******************************************************************************

  * Usage:
        getVariableToIndexset(R, indexTuple)
  * Inputs:
      * R, a ring (Macaulay2Doc), A polynomial ring whose variables
        are indexed by tuples.
      * indexTuple, a list (Macaulay2Doc), A list representing the
        index of the variable.
  * Outputs:
      * a ring element (Macaulay2Doc), The variable from the
        ring corresponding to the given index.

Description
===========

Given a ring $R = \mathbb{Q}[p_{\{i,j\}}]$, this method retrieves $p_{\{i,j\}}$
when passed the index $\{i,j\}$.

+-----------------------------------------------+
|i1 : R = QQ[p_{0,0}, p_{0,1}, p_{1,0}, p_{1,1}]|
|                                               |
|o1 = R                                         |
|                                               |
|o1 : PolynomialRing                            |
+-----------------------------------------------+
|i2 : getVariableToIndexset(R, {1,0})           |
|                                               |
|o2 = p                                         |
|      {1, 0}                                   |
|                                               |
|o2 : R                                         |
+-----------------------------------------------+

See also
========

  * probabilityRing -- ring of probability
    distributions of a game indexed by ordered multi-indices
  * assemblePolynomial -- compute incentive
    constraint polynomial for deviation

Ways to use getVariableToIndexset:
==================================

  * "getVariableToIndexset(Ring,List)"

For the programmer
==================

The object getVariableToIndexset, is a
method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1178:0.

indexset -- get the list of index tuples of a tensor
****************************************************

  * Usage:
        indexset T
  * Inputs:
      * T, an instance of the type Tensor, A tensor whose index
        set is to be retrieved.
  * Outputs:
      * a list (Macaulay2Doc), A list of index tuples corresponding
        to the entries of the tensor.

Description
===========

Given a tensor $T$ of format $\{d_1, d_2, \dots, d_n\}$, this method returns a
list of all index tuples $(i_1, i_2, \dots, i_n)$ such that $0 \leq i_j < d_j$
for each dimension $j$. Difference from enumerateTensorIndices,
is that indexset can only be used with a Tensor object.
See indexset, retrieves precomputed indices stored as metadata in a
Tensor.

+-----------------------------------------------------------------------------+
|i1 : T = zeroTensor(QQ, {2,3,4});                                            |
+-----------------------------------------------------------------------------+
|i2 : indexset T                                                              |
|                                                                             |
|o2 = {{0, 0, 0}, {0, 0, 1}, {0, 0, 2}, {0, 0, 3}, {0, 1, 0}, {0, 1, 1}, {0,  |
|     ------------------------------------------------------------------------|
|     1, 2}, {0, 1, 3}, {0, 2, 0}, {0, 2, 1}, {0, 2, 2}, {0, 2, 3}, {1, 0, 0},|
|     ------------------------------------------------------------------------|
|     {1, 0, 1}, {1, 0, 2}, {1, 0, 3}, {1, 1, 0}, {1, 1, 1}, {1, 1, 2}, {1, 1,|
|     ------------------------------------------------------------------------|
|     3}, {1, 2, 0}, {1, 2, 1}, {1, 2, 2}, {1, 2, 3}}                         |
|                                                                             |
|o2 : List                                                                    |
+-----------------------------------------------------------------------------+
+-----------------------------+
|i3 : T = zeroTensor(QQ, {2});|
+-----------------------------+
|i4 : indexset T              |
|                             |
|o4 = {{0}, {1}}              |
|                             |
|o4 : List                    |
+-----------------------------+

See also
========

  * zeroTensor -- construct a tensor with zero entries from
    a given ring.
  * randomTensor -- construct a tensor with random entries
    from a given ring
  * enumerateTensorIndices -- generate index
    tuples for a tensor with given dimensions

Ways to use indexset:
=====================

  * "indexset(Tensor)"

For the programmer
==================

The object indexset, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1023:0.

intersectWithCImodel -- ideal of the intersection of a given variety with the conditional independence model
************************************************************************************************************

  * Usage:
        intersectWithCImodel(V, Stmts)
        intersectWithCImodel(V, Stmts, PlayerNames)
        intersectWithCImodel(V, G)
        intersectWithCImodel(V, G, PlayerNames)
  * Inputs:
      * V, an ideal (Macaulay2Doc), an ideal of a ring created
        with probabilityRing
      * Stmts, a list (Macaulay2Doc), the list of conditional
        independence statements
      * G, a graph (Graphs), the graph modelling the conditional
        dependencies between players
      * PlayerNames, a list (Macaulay2Doc), the ordered list of
        players - the names of the random variables in the conditional
        independence statements or vertices of the graph. If PlayerNames is
        omitted, the players (or the vertices of G) are assumed to be labelled
        1..n.
  * Optional inputs (Macaulay2Doc),:
      * Verbose (missing documentation) => ..., default value false,
  * Outputs:
      * an ideal (Macaulay2Doc), The ideal of the intersection of
        the given variety with the conditional independence model determined by
        the conditional independence statements/graph.

Description
===========

intersectWithCImodel calculates the ideal of the intersection of the given
variety V with the conditional independence model determined by a set of
conditional probability statements or an undirected graph. More precisely, the
output is the ideal of the closure of the variety given by removing the
components in the coordinate hyperplanes from the intersection of the variety V
and the conditional independence model.

The input for the conditional independence model can be a set of conditional
probability statements or an undirected graph. A single conditional
independence statement is a list consisting of three disjoint lists of indices
for random variables, e.g. $\{ \{1,2\},\{4\}, \{3\} \}$ which represents the
conditional independence statement ``$(X_1, X_2)$ is conditionally independent
of $X_4$ given $X_3$''. In the context of game theory, the variable $X_i$
represents the strategy of player $i$.

Given an undirected graph $G$, the conditional independence statements are
produced via the globalMarkov function from the GraphicalModels package. A
global Markov statement for $G$ is a list $\{A, B, C\}$ of three disjoint lists
of vertices of $G$, where the subset $C$ separates the subset $A$ from the
subset $B$ in the graph $G$.

+-----------------------------------------------------------------------------+
|i1 : FF = ZZ/32003                                                           |
|                                                                             |
|o1 = FF                                                                      |
|                                                                             |
|o1 : QuotientRing                                                            |
+-----------------------------------------------------------------------------+
|i2 : d = {2,2,2};                                                            |
+-----------------------------------------------------------------------------+
|i3 : X = randomGame(d, CoefficientRing => FF);                               |
+-----------------------------------------------------------------------------+
|i4 : PR = probabilityRing(d, CoefficientRing => FF);                         |
+-----------------------------------------------------------------------------+
|i5 : V = spohnIdeal(PR, X);                                                  |
|                                                                             |
|o5 : Ideal of PR                                                             |
+-----------------------------------------------------------------------------+
|i6 : G1 = graph ({}, Singletons => {1,2,3});                                 |
+-----------------------------------------------------------------------------+
|i7 : G2 = graph ({{1,2}}, Singletons => {3});                                |
+-----------------------------------------------------------------------------+
|i8 : I1 = intersectWithCImodel(V, G1)                                        |
|                                                                             |
|                                                                             |
|o8 = ideal (p          + 3034p          - 14745p         , p          +      |
|             {1, 0, 1}        {1, 1, 0}         {1, 1, 1}   {1, 0, 0}        |
|     ------------------------------------------------------------------------|
|                                                                             |
|     735p          - 10307p         , p          - 9548p          - 11660p   |
|         {1, 1, 0}         {1, 1, 1}   {0, 1, 1}        {1, 1, 0}         {1,|
|     ------------------------------------------------------------------------|
|                                                                             |
|          , p          + 6605p          - 4419p         , p          -       |
|     1, 1}   {0, 1, 0}        {1, 1, 0}        {1, 1, 1}   {0, 0, 1}         |
|     ------------------------------------------------------------------------|
|                                                                             |
|     9755p          - 8595p         , p          + 7770p          + 7307p    |
|          {1, 1, 0}        {1, 1, 1}   {0, 0, 0}        {1, 1, 0}        {1, |
|     ------------------------------------------------------------------------|
|             2                                            2                  |
|          , p          + 15142p         p          - 9838p         )         |
|     1, 1}   {1, 1, 0}         {1, 1, 0} {1, 1, 1}        {1, 1, 1}          |
|                                                                             |
|o8 : Ideal of PR                                                             |
+-----------------------------------------------------------------------------+
|i9 : I2 = intersectWithCImodel(V, G2)                                        |
|                                                                             |
|                                                                             |
|o9 = ideal (p          + 4385p          + 10227p          + 4120p         ,  |
|             {0, 0, 1}        {0, 1, 1}         {1, 0, 1}        {1, 1, 1}   |
|     ------------------------------------------------------------------------|
|                                                                             |
|     p          + 4385p          + 10227p          + 4120p         , p       |
|      {0, 0, 0}        {0, 1, 0}         {1, 0, 0}        {1, 1, 0}   {1, 0, |
|     ------------------------------------------------------------------------|
|                                                                             |
|       p          - p         p         , p         p          - p           |
|     1} {1, 1, 0}    {1, 0, 0} {1, 1, 1}   {0, 1, 1} {1, 1, 0}    {0, 1,     |
|     ------------------------------------------------------------------------|
|                                                                             |
|       p         , p         p          + 848p         p          + 12103p   |
|     0} {1, 1, 1}   {0, 1, 0} {1, 0, 1}       {0, 1, 1} {1, 0, 1}         {1,|
|     ------------------------------------------------------------------------|
|                            2                                                |
|          p          - 6085p          + 7894p         p          - 3300p     |
|     0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1, 0} {1, 1, 1}        {0,  |
|     ------------------------------------------------------------------------|
|                                                                             |
|          p          - 850p         p          - 10956p         p          + |
|     1, 1} {1, 1, 1}       {1, 0, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}   |
|     ------------------------------------------------------------------------|
|                                     2                                       |
|     6865p         p          + 8470p         , p         p          +       |
|          {1, 1, 0} {1, 1, 1}        {1, 1, 1}   {0, 1, 1} {1, 0, 0}         |
|     ------------------------------------------------------------------------|
|                                                                2            |
|     848p         p          + 12103p         p          - 6085p          +  |
|         {0, 1, 1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}        {1, 0, 1}    |
|     ------------------------------------------------------------------------|
|                                                                             |
|     7894p         p          - 3300p         p          - 850p         p    |
|          {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}       {1, 0, 0} {1, |
|     ------------------------------------------------------------------------|
|                                                                         2   |
|           - 10956p         p          + 6865p         p          + 8470p    |
|     1, 1}         {1, 0, 1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}        {1, |
|     ------------------------------------------------------------------------|
|                                         2                                   |
|          , p         p          + 12103p          - 15038p         p        |
|     1, 1}   {0, 1, 0} {1, 0, 0}         {1, 0, 0}         {0, 1, 1} {1, 0,  |
|     ------------------------------------------------------------------------|
|                                          2                                  |
|        + 3534p         p          + 7597p          + 7894p         p        |
|     1}        {1, 0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1, 0} {1, 1,  |
|     ------------------------------------------------------------------------|
|                                         2                                   |
|        - 850p         p          + 6865p          - 8785p         p         |
|     0}       {1, 0, 0} {1, 1, 0}        {1, 1, 0}        {0, 1, 0} {1, 1, 1}|
|     ------------------------------------------------------------------------|
|                                                                             |
|     + 14139p         p          + 5778p         p          + 9818p          |
|             {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {1, 0,    |
|     ------------------------------------------------------------------------|
|                                                      2                      |
|       p          + 11496p         p          - 13888p         , p           |
|     1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}         {1, 1, 1}   {0, 1,     |
|     ------------------------------------------------------------------------|
|                         2                                                   |
|       p          + 4731p          - 15131p         p          + 11961p      |
|     0} {0, 1, 1}        {0, 1, 1}         {0, 1, 1} {1, 0, 1}         {1, 0,|
|     ------------------------------------------------------------------------|
|                          2                                                  |
|       p          - 13703p          - 11806p         p          - 7596p      |
|     0} {1, 0, 1}         {1, 0, 1}         {0, 1, 0} {1, 1, 1}        {0, 1,|
|     ------------------------------------------------------------------------|
|                                                                             |
|       p          + 10561p         p          + 8010p         p          -   |
|     1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}     |
|     ------------------------------------------------------------------------|
|                                      2           2                 2        |
|     10049p         p          + 1546p         , p          - 12264p         |
|           {1, 1, 0} {1, 1, 1}        {1, 1, 1}   {0, 1, 0}         {0, 1, 1}|
|     ------------------------------------------------------------------------|
|             2                                                               |
|     + 11961p          - 8065p         p          - 10563p         p         |
|             {1, 0, 0}        {0, 1, 1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}|
|     ------------------------------------------------------------------------|
|            2                                                                |
|     - 8689p          - 11806p         p          + 10561p         p         |
|            {1, 0, 1}         {0, 1, 0} {1, 1, 0}         {1, 0, 0} {1, 1, 0}|
|     ------------------------------------------------------------------------|
|             2                                                               |
|     - 10049p          + 10273p         p          - 10313p         p        |
|             {1, 1, 0}         {0, 1, 0} {1, 1, 1}         {0, 1, 1} {1, 1,  |
|     ------------------------------------------------------------------------|
|                                                                             |
|        + 4458p         p          - 3454p         p          + 11487p       |
|     1}        {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}         {1, 1, |
|     ------------------------------------------------------------------------|
|                         2                                                   |
|       p          + 2116p         )                                          |
|     0} {1, 1, 1}        {1, 1, 1}                                           |
|                                                                             |
|o9 : Ideal of PR                                                             |
+-----------------------------------------------------------------------------+

Here is an example where the vertices of the graph need to be relabeled.

+-----------------------------------------------------------------------------+
|i10 : FF = ZZ/32003;                                                         |
+-----------------------------------------------------------------------------+
|i11 : d = {2,2,2};                                                           |
+-----------------------------------------------------------------------------+
|i12 : X = randomGame(d, CoefficientRing => FF);                              |
+-----------------------------------------------------------------------------+
|i13 : PR = probabilityRing(d, CoefficientRing => FF);                        |
+-----------------------------------------------------------------------------+
|i14 : V = spohnIdeal(PR, X);                                                 |
|                                                                             |
|o14 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i15 : G1 = graph {{John,Matthew},{Matthew,Sarah}};                           |
+-----------------------------------------------------------------------------+
|i16 : G2 = graph {{a,b},{b,c},{c,a}};                                        |
+-----------------------------------------------------------------------------+
|i17 : I1 = intersectWithCImodel(V, G1, {John,Matthew,Sarah})                 |
|                                                                             |
|                                                                             |
|o17 = ideal (p         p          - p         p         , p         p        |
|              {0, 1, 1} {1, 1, 0}    {0, 1, 0} {1, 1, 1}   {0, 0, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|         - p         p         , p         p          - 7572p         p      |
|      0}    {0, 0, 0} {1, 0, 1}   {0, 0, 0} {1, 0, 0}        {0, 1, 0} {1, 0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|         - 6520p         p          - 4513p         p          - 4514p       |
|      0}        {0, 1, 1} {1, 0, 0}        {0, 0, 0} {1, 0, 1}        {0, 0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 9722p         p          - 8670p         p          -   |
|      1} {1, 0, 1}        {0, 1, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      8060p         p          - 10425p         p          - 15633p          |
|           {0, 0, 0} {1, 1, 0}         {0, 0, 1} {1, 1, 0}         {0, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 11623p         p          + 9258p         p          -  |
|      0} {1, 1, 0}         {0, 0, 0} {1, 1, 1}        {0, 0, 1} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                                                                             |
|      10531p         p          + 5102p         p         , p         p      |
|            {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}   {0, 0, 0} {0, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|         - 15219p         p          - 1624p         p          + 15159p     |
|      0}         {0, 0, 1} {0, 1, 0}        {0, 0, 0} {0, 1, 1}         {0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p          - 12583p         p          - 14208p         p         |
|      0, 1} {0, 1, 1}         {0, 1, 0} {1, 0, 0}         {0, 1, 1} {1, 0, 0}|
|      -----------------------------------------------------------------------|
|                                                                             |
|      + 11304p         p          + 9679p         p          + 9627p         |
|              {0, 1, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}        {0, 0,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 5593p         p          - 2957p         p          -   |
|      0} {1, 1, 0}        {0, 0, 1} {1, 1, 0}        {1, 0, 0} {1, 1, 0}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      11073p         p          - 6617p         p          + 10166p          |
|            {1, 0, 1} {1, 1, 0}        {0, 0, 0} {1, 1, 1}         {0, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 12802p         p          + 4686p         p         ,   |
|      1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                                                                             |
|      p         p          + 8469p         p          - 10163p         p     |
|       {0, 0, 0} {0, 0, 1}        {0, 0, 1} {0, 1, 0}         {0, 0, 0} {0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|            - 1695p         p          - 14677p         p          +         |
|      1, 1}        {0, 1, 0} {0, 1, 1}         {0, 1, 1} {1, 0, 0}           |
|      -----------------------------------------------------------------------|
|                                                                             |
|      11152p         p          - 7870p         p          + 11151p          |
|            {0, 0, 0} {1, 0, 1}        {0, 1, 0} {1, 0, 1}         {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 15920p         p          - 256p         p          +   |
|      0} {1, 0, 1}         {0, 0, 1} {1, 1, 0}       {1, 0, 1} {1, 1, 0}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      12005p         p          - 5611p         p          + 7491p           |
|            {0, 0, 0} {1, 1, 1}        {0, 1, 0} {1, 1, 1}        {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 3916p         p         , p         p         p         |
|      0} {1, 1, 1}        {1, 1, 0} {1, 1, 1}   {0, 1, 0} {1, 0, 0} {1, 0,   |
|      -----------------------------------------------------------------------|
|                          2                                              2   |
|        p          - 2025p         p         p          - 8855p         p    |
|      1} {1, 1, 0}        {1, 0, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 0} {1, |
|      -----------------------------------------------------------------------|
|                                        2                                    |
|           p          + 15298p         p         p          - 2246p          |
|      0, 1} {1, 1, 0}         {0, 0, 1} {1, 0, 1} {1, 1, 0}        {0, 1,    |
|      -----------------------------------------------------------------------|
|         2                                    2                              |
|        p         p          + 5120p         p         p          + 11477p   |
|      0} {1, 0, 1} {1, 1, 0}        {1, 0, 0} {1, 0, 1} {1, 1, 0}         {0,|
|      -----------------------------------------------------------------------|
|                      2                                    2                 |
|           p         p          + 7638p         p         p          +       |
|      0, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {1, 0, 1} {1, 1, 0}         |
|      -----------------------------------------------------------------------|
|                               2                                     2       |
|      1268p         p         p          - 14784p         p         p        |
|           {0, 1, 0} {1, 0, 1} {1, 1, 0}         {1, 0, 0} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                2         2               2                                  |
|         + 6758p         p          + 426p         p         p          +    |
|      0}        {1, 0, 1} {1, 1, 0}       {0, 1, 1} {1, 0, 0} {1, 1, 1}      |
|      -----------------------------------------------------------------------|
|                     2                                     2                 |
|      9922p         p         p          + 15161p         p         p        |
|           {0, 1, 0} {1, 0, 0} {1, 1, 1}         {0, 1, 1} {1, 0, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                2                                                            |
|         + 4579p         p         p          + 11132p         p         p   |
|      1}        {0, 0, 1} {1, 0, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|                             2                                               |
|           p          + 5195p         p         p          - 9636p           |
|      0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}        {0, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 10604p         p         p          |
|      0} {0, 1, 1} {1, 0, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 1} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                          2  |
|        p          + 7191p         p         p         p          - 14257p   |
|      1} {1, 1, 1}        {0, 1, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1}         {0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p          - 13121p         p         p         p       |
|      1, 1} {1, 0, 1} {1, 1, 1}         {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|                                                                2            |
|         + 12659p         p         p         p          + 7266p         p   |
|      1}         {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}        {1, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|                                       2                                     |
|           p          - 8855p         p         p          + 3518p           |
|      0, 1} {1, 1, 1}        {0, 0, 0} {1, 0, 1} {1, 1, 1}        {0, 0,     |
|      -----------------------------------------------------------------------|
|         2                                    2                              |
|        p         p          - 4926p         p         p          - 15176p   |
|      1} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2                                    2                           |
|           p         p          - 1233p         p         p          -       |
|      1, 1} {1, 0, 1} {1, 1, 1}        {1, 0, 0} {1, 0, 1} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|            2                                                                |
|      13140p         p         p          - 14057p         p         p       |
|            {0, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {0, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                          2                                                  |
|        p          + 9623p         p         p          + 7952p         p    |
|      0} {1, 1, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|                                       2                                     |
|           p         p          + 3226p         p         p          +       |
|      0, 0} {1, 1, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 0} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|                                                                             |
|      12897p         p         p         p          - 13896p         p       |
|            {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {1, 0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 694p         p         p         p          + |
|      1} {1, 1, 0} {1, 1, 1}       {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}   |
|      -----------------------------------------------------------------------|
|                                                           2                 |
|      11611p         p         p         p          - 6750p         p        |
|            {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                    2                                    2   |
|        p          + 9008p         p         p          - 8064p         p    |
|      0} {1, 1, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|                                       2                                     |
|           p          - 4313p         p         p          - 5828p           |
|      1, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 0} {1, 1, 1}        {1, 0,     |
|      -----------------------------------------------------------------------|
|         2                           2         2                             |
|        p         p          - 10710p         p          + 14163p            |
|      1} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {1, 1, 1}         {0, 0,      |
|      -----------------------------------------------------------------------|
|                   2               2         2                               |
|        p         p          + 790p         p          + 8184p         p     |
|      1} {0, 1, 0} {1, 1, 1}       {0, 1, 0} {1, 1, 1}        {0, 0, 1} {0,  |
|      -----------------------------------------------------------------------|
|            2                                   2                 2          |
|           p          - 228p         p         p          - 13861p           |
|      1, 1} {1, 1, 1}       {0, 1, 0} {0, 1, 1} {1, 1, 1}         {0, 1,     |
|      -----------------------------------------------------------------------|
|         2                                    2                              |
|        p          - 3536p         p         p          - 2039p         p    |
|      1} {1, 1, 1}        {0, 1, 0} {1, 0, 0} {1, 1, 1}        {0, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|            2                2         2                                     |
|           p          + 5602p         p          + 1594p         p           |
|      0, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {0, 0, 0} {1, 0,     |
|      -----------------------------------------------------------------------|
|         2                                    2                              |
|        p          - 1643p         p         p          + 13400p         p   |
|      1} {1, 1, 1}        {0, 0, 1} {1, 0, 1} {1, 1, 1}         {0, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|            2                                    2                           |
|           p          - 1626p         p         p          + 7660p           |
|      0, 1} {1, 1, 1}        {0, 1, 1} {1, 0, 1} {1, 1, 1}        {1, 0,     |
|      -----------------------------------------------------------------------|
|                   2                2         2                              |
|        p         p          - 6267p         p          + 11901p         p   |
|      0} {1, 0, 1} {1, 1, 1}        {1, 0, 1} {1, 1, 1}         {0, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|            2                                    2                           |
|           p          - 1563p         p         p          - 2002p           |
|      1, 0} {1, 1, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1}        {1, 0,     |
|      -----------------------------------------------------------------------|
|                   2                                     2                2  |
|        p         p          + 13912p         p         p          - 1383p   |
|      0} {1, 1, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 0} {1, 1, 1}        {1,|
|      -----------------------------------------------------------------------|
|            2                          3                           3         |
|           p          - 5666p         p          - 11175p         p          |
|      1, 0} {1, 1, 1}        {0, 0, 1} {1, 1, 1}         {0, 1, 0} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|                        3                          3                         |
|      + 12875p         p          + 5080p         p          - 7775p         |
|              {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {1, 0,   |
|      -----------------------------------------------------------------------|
|         3                          3                                        |
|        p          + 9606p         p         , p         p         p         |
|      1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}   {0, 0, 1} {0, 1, 0} {0, 1,   |
|      -----------------------------------------------------------------------|
|                           2                                                 |
|        p          - 14243p         p         p          - 15958p            |
|      1} {1, 0, 1}         {0, 1, 0} {0, 1, 1} {1, 0, 1}         {0, 0,      |
|      -----------------------------------------------------------------------|
|         2                                    2                              |
|        p         p          + 6166p         p         p          - 5634p    |
|      0} {0, 1, 1} {1, 0, 1}        {0, 0, 1} {0, 1, 1} {1, 0, 1}        {0, |
|      -----------------------------------------------------------------------|
|            2                           2                                    |
|           p         p          + 14492p         p         p          +      |
|      1, 0} {0, 1, 1} {1, 0, 1}         {0, 1, 1} {1, 0, 0} {1, 0, 1}        |
|      -----------------------------------------------------------------------|
|                     2                           2         2                 |
|      2501p         p         p          - 15165p         p          -       |
|           {0, 1, 1} {1, 0, 0} {1, 0, 1}         {0, 0, 1} {1, 0, 1}         |
|      -----------------------------------------------------------------------|
|                                2               2         2                  |
|      15250p         p         p          - 406p         p          +        |
|            {0, 0, 1} {0, 1, 0} {1, 0, 1}       {0, 1, 0} {1, 0, 1}          |
|      -----------------------------------------------------------------------|
|                               2                                     2       |
|      3752p         p         p          + 12382p         p         p        |
|           {0, 0, 0} {0, 1, 1} {1, 0, 1}         {0, 0, 1} {0, 1, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                     2                2         2            |
|         - 11217p         p         p          + 6170p         p          -  |
|      1}         {0, 1, 0} {0, 1, 1} {1, 0, 1}        {0, 1, 1} {1, 0, 1}    |
|      -----------------------------------------------------------------------|
|                               2                                    2        |
|      7630p         p         p          + 7038p         p         p         |
|           {0, 1, 0} {1, 0, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 0} {1, 0, 1}|
|      -----------------------------------------------------------------------|
|              2         2                           3                        |
|      + 15402p         p          - 12723p         p          + 2012p        |
|              {1, 0, 0} {1, 0, 1}         {0, 0, 0} {1, 0, 1}        {0, 0,  |
|      -----------------------------------------------------------------------|
|         3                          3                          3             |
|        p          - 6736p         p          + 8283p         p          -   |
|      1} {1, 0, 1}        {0, 1, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}     |
|      -----------------------------------------------------------------------|
|                     3                                                       |
|      3584p         p          - 3794p         p         p         p         |
|           {1, 0, 0} {1, 0, 1}        {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1, 0}|
|      -----------------------------------------------------------------------|
|             2                                     2                         |
|      - 3223p         p         p          - 13454p         p         p      |
|             {0, 1, 0} {1, 0, 1} {1, 1, 0}         {1, 0, 0} {1, 0, 1} {1, 1,|
|      -----------------------------------------------------------------------|
|                           2                                    2            |
|         + 10279p         p         p          - 7844p         p         p   |
|      0}         {0, 0, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                             2                                    2          |
|            - 3516p         p         p          + 8958p         p           |
|      1, 0}        {0, 1, 0} {1, 0, 1} {1, 1, 0}        {1, 0, 0} {1, 0,     |
|      -----------------------------------------------------------------------|
|                          3                           2         2            |
|        p          - 8268p         p          + 11120p         p          -  |
|      1} {1, 1, 0}        {1, 0, 1} {1, 1, 0}         {0, 0, 1} {1, 1, 0}    |
|      -----------------------------------------------------------------------|
|                              2                                    2         |
|      117p         p         p          + 7933p         p         p          |
|          {0, 0, 1} {0, 1, 0} {1, 1, 0}        {0, 0, 0} {1, 0, 1} {1, 1, 0} |
|      -----------------------------------------------------------------------|
|                                 2                                    2      |
|      - 4540p         p         p          + 4523p         p         p       |
|             {0, 0, 1} {1, 0, 1} {1, 1, 0}        {0, 1, 0} {1, 0, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|                                    2                 2         2            |
|         + 6609p         p         p          + 11413p         p          +  |
|      0}        {1, 0, 0} {1, 0, 1} {1, 1, 0}         {1, 0, 1} {1, 1, 0}    |
|      -----------------------------------------------------------------------|
|                      3                          3                           |
|      14051p         p          + 3378p         p          - 2516p           |
|            {0, 0, 1} {1, 1, 0}        {1, 0, 1} {1, 1, 0}        {0, 0,     |
|      -----------------------------------------------------------------------|
|         2                          3                                        |
|        p         p          + 3051p         p          - 6094p         p    |
|      1} {0, 1, 0} {1, 1, 1}        {0, 1, 0} {1, 1, 1}        {0, 0, 1} {0, |
|      -----------------------------------------------------------------------|
|                                       2                                     |
|           p         p          + 4590p         p         p          +       |
|      1, 0} {0, 1, 1} {1, 1, 1}        {0, 1, 0} {0, 1, 1} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|                      2                           2                          |
|      10665p         p         p          - 13935p         p         p       |
|            {0, 1, 0} {0, 1, 1} {1, 1, 1}         {0, 1, 1} {1, 0, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                          2                                    2             |
|         + 4295p         p         p          - 6110p         p         p    |
|      1}        {0, 1, 0} {1, 0, 0} {1, 1, 1}        {0, 1, 1} {1, 0, 0} {1, |
|      -----------------------------------------------------------------------|
|                   2                                                         |
|            - 9317p         p         p          + 8038p         p           |
|      1, 1}        {0, 0, 1} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {0, 1,     |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          - 8809p         p         p          - 5623p    |
|      0} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p          + 3537p         p         p        |
|      0, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {0, 1, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                          2  |
|        p          - 14684p         p         p         p          - 8102p   |
|      1} {1, 1, 1}         {0, 1, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1}        {0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p          + 15472p         p         p         p       |
|      1, 1} {1, 0, 1} {1, 1, 1}         {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|                                                                2            |
|         - 13852p         p         p         p          - 5666p         p   |
|      1}         {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}        {1, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|                                       2                                     |
|           p          - 2863p         p         p          - 14919p          |
|      0, 1} {1, 1, 1}        {0, 0, 0} {1, 0, 1} {1, 1, 1}         {0, 0,    |
|      -----------------------------------------------------------------------|
|         2                                    2                              |
|        p         p          + 1694p         p         p          - 360p     |
|      1} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}       {0,  |
|      -----------------------------------------------------------------------|
|            2                                    2                           |
|           p         p          - 9597p         p         p          +       |
|      1, 1} {1, 0, 1} {1, 1, 1}        {1, 0, 0} {1, 0, 1} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|            3                           2                                    |
|      11132p         p          + 10999p         p         p          -      |
|            {1, 0, 1} {1, 1, 1}         {0, 0, 1} {1, 1, 0} {1, 1, 1}        |
|      -----------------------------------------------------------------------|
|                                                          2                  |
|      13375p         p         p         p          + 121p         p         |
|            {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 1, 1}       {0, 1, 0} {1, 1,   |
|      -----------------------------------------------------------------------|
|                                                                         2   |
|        p          - 7302p         p         p         p          - 6621p    |
|      0} {1, 1, 1}        {0, 1, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1}        {1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p          + 16p         p         p         p          |
|      0, 0} {1, 1, 0} {1, 1, 1}      {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      - 14517p         p         p         p          - 14891p         p     |
|              {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 1, 0} {1,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p          + 10497p         p         p         p       |
|      0, 1} {1, 1, 0} {1, 1, 1}         {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                 2                                               2           |
|         - 13928p         p         p          - 14933p         p            |
|      1}         {1, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {1, 1,      |
|      -----------------------------------------------------------------------|
|                                    2                                    2   |
|        p          + 9570p         p         p          - 2423p         p    |
|      0} {1, 1, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1}        {1, 0, 0} {1, |
|      -----------------------------------------------------------------------|
|                                        2                           3        |
|           p          - 11148p         p         p          + 10669p         |
|      1, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 0} {1, 1, 1}         {1, 1,   |
|      -----------------------------------------------------------------------|
|                           2         2                                    2  |
|        p          + 14252p         p          + 4352p         p         p   |
|      0} {1, 1, 1}         {0, 0, 1} {1, 1, 1}        {0, 0, 1} {0, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|                   2         2                                    2          |
|            - 5405p         p          - 2494p         p         p          -|
|      1, 1}        {0, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 1, 1} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                               2               2         2                   |
|      2458p         p         p          + 221p         p          - 8810p   |
|           {0, 1, 0} {0, 1, 1} {1, 1, 1}       {0, 1, 1} {1, 1, 1}        {0,|
|      -----------------------------------------------------------------------|
|                      2                                     2                |
|           p         p          - 13968p         p         p          +      |
|      1, 0} {1, 0, 0} {1, 1, 1}         {0, 1, 1} {1, 0, 0} {1, 1, 1}        |
|      -----------------------------------------------------------------------|
|           2         2                                    2                  |
|      8726p         p          - 7769p         p         p          -        |
|           {1, 0, 0} {1, 1, 1}        {0, 0, 0} {1, 0, 1} {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|                               2                                    2        |
|      2338p         p         p          + 3358p         p         p         |
|           {0, 0, 1} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}|
|      -----------------------------------------------------------------------|
|                                  2                                    2     |
|      - 11821p         p         p          + 2532p         p         p      |
|              {0, 1, 1} {1, 0, 1} {1, 1, 1}        {1, 0, 0} {1, 0, 1} {1, 1,|
|      -----------------------------------------------------------------------|
|                 2         2                                    2            |
|         - 12316p         p          + 5849p         p         p          +  |
|      1}         {1, 0, 1} {1, 1, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                                2                                    2       |
|      15371p         p         p          + 9917p         p         p        |
|            {0, 1, 0} {1, 1, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                    2               2         2              |
|         + 8415p         p         p          - 784p         p          -    |
|      1}        {1, 0, 1} {1, 1, 0} {1, 1, 1}       {1, 1, 0} {1, 1, 1}      |
|      -----------------------------------------------------------------------|
|                     3                          3                            |
|      6420p         p          + 3362p         p          + 13997p           |
|           {0, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 1, 1}         {0, 1,     |
|      -----------------------------------------------------------------------|
|         3                          3                           3            |
|        p          - 9431p         p          + 12750p         p          +  |
|      1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                      3           2                                    2     |
|      11941p         p         , p         p         p          + 8301p      |
|            {1, 1, 0} {1, 1, 1}   {0, 0, 1} {0, 1, 1} {1, 0, 1}        {0, 1,|
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 2872p         p         p          + 1676p    |
|      0} {0, 1, 1} {1, 0, 1}        {0, 0, 0} {0, 1, 1} {1, 0, 1}        {0, |
|      -----------------------------------------------------------------------|
|            2                                    2                           |
|           p         p          + 8028p         p         p          -       |
|      0, 1} {0, 1, 1} {1, 0, 1}        {0, 1, 0} {0, 1, 1} {1, 0, 1}         |
|      -----------------------------------------------------------------------|
|           2                                              2                  |
|      9021p         p         p          - 1907p         p         p         |
|           {0, 1, 1} {1, 0, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 0} {1, 0, 1}|
|      -----------------------------------------------------------------------|
|             2         2                                    2                |
|      + 5805p         p          + 4603p         p         p          +      |
|             {0, 0, 1} {1, 0, 1}        {0, 0, 1} {0, 1, 0} {1, 0, 1}        |
|      -----------------------------------------------------------------------|
|            2         2                                    2                 |
|      14590p         p          + 6650p         p         p          +       |
|            {0, 1, 0} {1, 0, 1}        {0, 0, 0} {0, 1, 1} {1, 0, 1}         |
|      -----------------------------------------------------------------------|
|                              2                                     2        |
|      101p         p         p          + 11496p         p         p         |
|          {0, 0, 1} {0, 1, 1} {1, 0, 1}         {0, 1, 0} {0, 1, 1} {1, 0, 1}|
|      -----------------------------------------------------------------------|
|             2         2                                   2                 |
|      + 9211p         p          - 593p         p         p          +       |
|             {0, 1, 1} {1, 0, 1}       {0, 1, 0} {1, 0, 0} {1, 0, 1}         |
|      -----------------------------------------------------------------------|
|                                2               2         2                  |
|      11858p         p         p          + 182p         p          -        |
|            {0, 1, 1} {1, 0, 0} {1, 0, 1}       {1, 0, 0} {1, 0, 1}          |
|      -----------------------------------------------------------------------|
|                     3                          3                            |
|      7855p         p          - 6563p         p          - 1505p            |
|           {0, 0, 0} {1, 0, 1}        {0, 0, 1} {1, 0, 1}        {0, 1,      |
|      -----------------------------------------------------------------------|
|         3                          3                           3            |
|        p          - 2221p         p          + 11818p         p          -  |
|      0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}    |
|      -----------------------------------------------------------------------|
|                                                            2                |
|      13742p         p         p         p          - 12821p         p       |
|            {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1, 0}         {0, 1, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|                          2                                              2   |
|        p          - 5391p         p         p          + 7556p         p    |
|      1} {1, 1, 0}        {1, 0, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 0} {1, |
|      -----------------------------------------------------------------------|
|                                       2                                     |
|           p          + 9935p         p         p          + 6379p           |
|      0, 1} {1, 1, 0}        {0, 0, 1} {1, 0, 1} {1, 1, 0}        {0, 1,     |
|      -----------------------------------------------------------------------|
|         2                                     2                          3  |
|        p         p          - 11501p         p         p          - 5097p   |
|      0} {1, 0, 1} {1, 1, 0}         {1, 0, 0} {1, 0, 1} {1, 1, 0}        {1,|
|      -----------------------------------------------------------------------|
|                             2         2                                     |
|           p          + 4765p         p          - 525p         p            |
|      0, 1} {1, 1, 0}        {0, 0, 1} {1, 1, 0}       {0, 0, 1} {0, 1,      |
|      -----------------------------------------------------------------------|
|         2                                     2                             |
|        p          - 12818p         p         p          - 3960p         p   |
|      0} {1, 1, 0}         {0, 0, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|            2                                    2                           |
|           p          - 2681p         p         p          - 4809p           |
|      0, 1} {1, 1, 0}        {0, 1, 0} {1, 0, 1} {1, 1, 0}        {1, 0,     |
|      -----------------------------------------------------------------------|
|                   2                 2         2                          3  |
|        p         p          - 14435p         p          - 4239p         p   |
|      0} {1, 0, 1} {1, 1, 0}         {1, 0, 1} {1, 1, 0}        {0, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                             3                         2                     |
|            - 9460p         p          + 830p         p         p          - |
|      1, 0}        {1, 0, 1} {1, 1, 0}       {0, 0, 1} {0, 1, 0} {1, 1, 1}   |
|      -----------------------------------------------------------------------|
|           3                          2                                      |
|      3542p         p          + 9178p         p         p          +        |
|           {0, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 1, 1} {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|                                                           2                 |
|      13401p         p         p         p          + 8812p         p        |
|            {0, 0, 1} {0, 1, 0} {0, 1, 1} {1, 1, 1}        {0, 1, 0} {0, 1,  |
|      -----------------------------------------------------------------------|
|                                    2                                   2    |
|        p          - 3781p         p         p          + 754p         p     |
|      1} {1, 1, 1}        {0, 0, 1} {0, 1, 1} {1, 1, 1}       {0, 1, 0} {0,  |
|      -----------------------------------------------------------------------|
|                            2                                                |
|           p          - 939p         p         p          + 6143p            |
|      1, 1} {1, 1, 1}       {0, 1, 1} {1, 0, 0} {1, 1, 1}        {0, 1,      |
|      -----------------------------------------------------------------------|
|         2                                    2                          2   |
|        p         p          - 9628p         p         p          + 1145p    |
|      0} {1, 0, 0} {1, 1, 1}        {0, 1, 1} {1, 0, 0} {1, 1, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p          - 6841p         p         p         p        |
|      0, 1} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                 2                                                           |
|         - 12922p         p         p          + 5777p         p         p   |
|      1}         {0, 1, 0} {1, 0, 1} {1, 1, 1}        {0, 0, 0} {0, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p          - 13310p         p         p         p          -      |
|      0, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 1, 1}        |
|      -----------------------------------------------------------------------|
|                                                           2                 |
|      11476p         p         p         p          - 9476p         p        |
|            {0, 1, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1}        {0, 1, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 10793p         p         p         p          - 3645p   |
|      1} {1, 1, 1}         {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1, 1}        {0,|
|      -----------------------------------------------------------------------|
|                                                  2                          |
|           p         p         p          + 14904p         p         p       |
|      1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}         {1, 0, 0} {1, 0, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|                          2                                    2             |
|         + 1585p         p         p          + 7652p         p         p    |
|      1}        {0, 0, 0} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {1, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|                             2                                     2         |
|            + 3086p         p         p          + 14919p         p          |
|      1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}         {0, 1, 1} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                    2                          3             |
|        p          - 6168p         p         p          - 7490p         p    |
|      1} {1, 1, 1}        {1, 0, 0} {1, 0, 1} {1, 1, 1}        {1, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|                   2                                                         |
|            + 3678p         p         p          - 9667p         p           |
|      1, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 1,     |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          - 3560p         p         p          - 15533p   |
|      0} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1}         {0,|
|      -----------------------------------------------------------------------|
|                                                  2                          |
|           p         p         p          - 14939p         p         p       |
|      1, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1}         {1, 0, 0} {1, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|         + 12326p         p         p         p          - 8128p         p   |
|      1}         {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p          - 4053p         p         p         p        |
|      0, 1} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                                                2            |
|         + 14913p         p         p         p          + 2800p         p   |
|      1}         {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                                       2                                     |
|           p          - 5138p         p         p          + 3554p           |
|      1, 0} {1, 1, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 1,     |
|      -----------------------------------------------------------------------|
|         2                                    2                              |
|        p         p          + 2257p         p         p          - 10618p   |
|      0} {1, 1, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 0} {1, 1, 1}         {1,|
|      -----------------------------------------------------------------------|
|            2                          3                          2          |
|           p         p          + 9306p         p          - 3609p           |
|      0, 1} {1, 1, 0} {1, 1, 1}        {1, 1, 0} {1, 1, 1}        {0, 0,     |
|      -----------------------------------------------------------------------|
|         2                                     2                 2           |
|        p          - 12092p         p         p          - 12825p            |
|      1} {1, 1, 1}         {0, 0, 1} {0, 1, 0} {1, 1, 1}         {0, 1,      |
|      -----------------------------------------------------------------------|
|         2                                     2                             |
|        p          + 14746p         p         p          + 12149p            |
|      0} {1, 1, 1}         {0, 0, 1} {0, 1, 1} {1, 1, 1}         {0, 1,      |
|      -----------------------------------------------------------------------|
|                   2               2         2                               |
|        p         p          - 615p         p          + 13952p         p    |
|      0} {0, 1, 1} {1, 1, 1}       {0, 1, 1} {1, 1, 1}         {0, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|            2                                    2                2          |
|           p          - 2536p         p         p          + 2032p           |
|      0, 0} {1, 1, 1}        {0, 1, 1} {1, 0, 0} {1, 1, 1}        {1, 0,     |
|      -----------------------------------------------------------------------|
|         2                                    2                              |
|        p          + 4764p         p         p          - 6395p         p    |
|      0} {1, 1, 1}        {0, 0, 0} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|            2                                    2                           |
|           p          - 9950p         p         p          - 6007p           |
|      0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}        {0, 1,     |
|      -----------------------------------------------------------------------|
|                   2                                     2                2  |
|        p         p          + 11585p         p         p          + 8863p   |
|      1} {1, 0, 1} {1, 1, 1}         {1, 0, 0} {1, 0, 1} {1, 1, 1}        {1,|
|      -----------------------------------------------------------------------|
|            2                                    2                           |
|           p          + 6661p         p         p          + 14790p          |
|      0, 1} {1, 1, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 1,    |
|      -----------------------------------------------------------------------|
|                   2                                     2                   |
|        p         p          - 13623p         p         p          -         |
|      0} {1, 1, 0} {1, 1, 1}         {1, 0, 0} {1, 1, 0} {1, 1, 1}           |
|      -----------------------------------------------------------------------|
|                                2                2         2                 |
|      11140p         p         p          - 6313p         p          +       |
|            {1, 0, 1} {1, 1, 0} {1, 1, 1}        {1, 1, 0} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|                     3                           3                           |
|      7416p         p          + 15703p         p          + 4551p           |
|           {0, 0, 1} {1, 1, 1}         {0, 1, 0} {1, 1, 1}        {0, 1,     |
|      -----------------------------------------------------------------------|
|         3                           3                          3            |
|        p          + 12129p         p          - 6987p         p          +  |
|      1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                     3                                                       |
|      1926p         p         )                                              |
|           {1, 1, 0} {1, 1, 1}                                               |
|                                                                             |
|o17 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i18 : I2 = intersectWithCImodel(V, G2, {a,b,c})                              |
|                                                                             |
|o18 = ideal (p         p          - 2364p         p          - 7572p         |
|              {0, 0, 0} {1, 0, 0}        {0, 0, 1} {1, 0, 0}        {0, 1,   |
|      -----------------------------------------------------------------------|
|        p          - 6520p         p          - 2149p         p          -   |
|      0} {1, 0, 0}        {0, 1, 1} {1, 0, 0}        {0, 0, 0} {1, 0, 1}     |
|      -----------------------------------------------------------------------|
|      4514p         p          - 9722p         p          - 8670p            |
|           {0, 0, 1} {1, 0, 1}        {0, 1, 0} {1, 0, 1}        {0, 1,      |
|      -----------------------------------------------------------------------|
|        p          - 8060p         p          - 10425p         p          -  |
|      1} {1, 0, 1}        {0, 0, 0} {1, 1, 0}         {0, 0, 1} {1, 1, 0}    |
|      -----------------------------------------------------------------------|
|      15633p         p          - 14581p         p          + 11623p         |
|            {0, 1, 0} {1, 1, 0}         {0, 1, 1} {1, 1, 0}         {0, 0,   |
|      -----------------------------------------------------------------------|
|        p          + 9258p         p          + 4050p         p          +   |
|      0} {1, 1, 1}        {0, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 1, 1}     |
|      -----------------------------------------------------------------------|
|      5102p         p         , p         p          - 15219p         p      |
|           {0, 1, 1} {1, 1, 1}   {0, 0, 0} {0, 1, 0}         {0, 0, 1} {0, 1,|
|      -----------------------------------------------------------------------|
|         - 1624p         p          + 15159p         p          - 12583p     |
|      0}        {0, 0, 0} {0, 1, 1}         {0, 0, 1} {0, 1, 1}         {0,  |
|      -----------------------------------------------------------------------|
|           p          - 14208p         p          + 11304p         p         |
|      1, 0} {1, 0, 0}         {0, 1, 1} {1, 0, 0}         {0, 1, 0} {1, 0, 1}|
|      -----------------------------------------------------------------------|
|      + 9679p         p          + 9627p         p          - 5593p          |
|             {0, 1, 1} {1, 0, 1}        {0, 0, 0} {1, 1, 0}        {0, 0,    |
|      -----------------------------------------------------------------------|
|        p          - 2957p         p          - 11073p         p          -  |
|      1} {1, 1, 0}        {1, 0, 0} {1, 1, 0}         {1, 0, 1} {1, 1, 0}    |
|      -----------------------------------------------------------------------|
|      6617p         p          + 10166p         p          + 12802p          |
|           {0, 0, 0} {1, 1, 1}         {0, 0, 1} {1, 1, 1}         {1, 0,    |
|      -----------------------------------------------------------------------|
|        p          + 4686p         p         , p         p          +        |
|      0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}   {0, 0, 0} {0, 0, 1}          |
|      -----------------------------------------------------------------------|
|      8469p         p          - 10163p         p          - 1695p           |
|           {0, 0, 1} {0, 1, 0}         {0, 0, 0} {0, 1, 1}        {0, 1,     |
|      -----------------------------------------------------------------------|
|        p          - 4513p         p          - 14677p         p          +  |
|      0} {0, 1, 1}        {0, 0, 1} {1, 0, 0}         {0, 1, 1} {1, 0, 0}    |
|      -----------------------------------------------------------------------|
|      15665p         p          - 7870p         p          + 11151p          |
|            {0, 0, 0} {1, 0, 1}        {0, 1, 0} {1, 0, 1}         {1, 0,    |
|      -----------------------------------------------------------------------|
|        p          - 15920p         p          + 5919p         p          -  |
|      0} {1, 0, 1}         {0, 0, 1} {1, 1, 0}        {0, 1, 1} {1, 1, 0}    |
|      -----------------------------------------------------------------------|
|      256p         p          + 12005p         p          - 11530p           |
|          {1, 0, 1} {1, 1, 0}         {0, 0, 0} {1, 1, 1}         {0, 1,     |
|      -----------------------------------------------------------------------|
|        p          + 7491p         p          - 3916p         p         )    |
|      0} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {1, 1, 0} {1, 1, 1}     |
|                                                                             |
|o18 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+

Here is an example where the conditional independence relations are given with
a List.

+-----------------------------------------------------------------------------+
|i19 : FF = ZZ/32003;                                                         |
+-----------------------------------------------------------------------------+
|i20 : d = {2,2,2};                                                           |
+-----------------------------------------------------------------------------+
|i21 : X = randomGame(d, CoefficientRing => FF);                              |
+-----------------------------------------------------------------------------+
|i22 : PR = probabilityRing(d, CoefficientRing => FF);                        |
+-----------------------------------------------------------------------------+
|i23 : V = spohnIdeal(PR, X);                                                 |
|                                                                             |
|o23 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i24 : G = graph ({{1,2}},Singletons => {3});                                 |
+-----------------------------------------------------------------------------+
|i25 : L = {{{1,2},{3},{}}};                                                  |
+-----------------------------------------------------------------------------+
|i26 : I1 = intersectWithCImodel(V, G)                                        |
|                                                                             |
|                                                                             |
|o26 = ideal (p          - 11083p          - 13237p          - 771p         , |
|              {0, 0, 1}         {0, 1, 1}         {1, 0, 1}       {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      p          - 11083p          - 13237p          - 771p         , p      |
|       {0, 0, 0}         {0, 1, 0}         {1, 0, 0}       {1, 1, 0}   {1, 0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - p         p         , p         p          - p          |
|      1} {1, 1, 0}    {1, 0, 0} {1, 1, 1}   {0, 1, 1} {1, 1, 0}    {0, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         , p         p          + 705p         p          + 9353p   |
|      0} {1, 1, 1}   {0, 1, 0} {1, 0, 1}       {0, 1, 1} {1, 0, 1}        {1,|
|      -----------------------------------------------------------------------|
|                             2                                               |
|           p          - 8959p          + 5062p         p          - 1880p    |
|      0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1, 0} {1, 1, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p          + 15763p         p          + 11084p         p         |
|      1, 1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}|
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      - 12581p         p          - 13990p         , p         p          +  |
|              {1, 1, 0} {1, 1, 1}         {1, 1, 1}   {0, 1, 1} {1, 0, 0}    |
|      -----------------------------------------------------------------------|
|                                                                2            |
|      705p         p          + 9353p         p          - 8959p          +  |
|          {0, 1, 1} {1, 0, 1}        {1, 0, 0} {1, 0, 1}        {1, 0, 1}    |
|      -----------------------------------------------------------------------|
|                                                                             |
|      5062p         p          - 1880p         p          + 15763p           |
|           {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}         {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 11084p         p          - 12581p         p          - |
|      0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}   |
|      -----------------------------------------------------------------------|
|            2                                      2                         |
|      13990p         , p         p          + 9353p          + 15023p        |
|            {1, 1, 1}   {0, 1, 0} {1, 0, 0}        {1, 0, 0}         {0, 1,  |
|      -----------------------------------------------------------------------|
|                                                       2                     |
|        p          - 10206p         p          + 11504p          + 5062p     |
|      1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}         {1, 0, 1}        {0,  |
|      -----------------------------------------------------------------------|
|                                                          2                  |
|           p          + 15763p         p          - 12581p          +        |
|      1, 0} {1, 1, 0}         {1, 0, 0} {1, 1, 0}         {1, 1, 0}          |
|      -----------------------------------------------------------------------|
|                                                                             |
|      13746p         p          + 13277p         p          + 3210p          |
|            {0, 1, 0} {1, 1, 1}         {0, 1, 1} {1, 1, 1}        {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 5488p         p          - 9216p         p          +   |
|      0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}     |
|      -----------------------------------------------------------------------|
|           2                                      2                          |
|      6026p         , p         p          - 7547p          + 13612p         |
|           {1, 1, 1}   {0, 1, 0} {0, 1, 1}        {0, 1, 1}         {0, 1,   |
|      -----------------------------------------------------------------------|
|                                                      2                      |
|        p          + 11912p         p          + 2824p          + 5387p      |
|      1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 13671p         p          - 4486p         p          -  |
|      0} {1, 1, 1}         {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                                                                  2          |
|      8494p         p          + 15671p         p          - 6785p         , |
|           {1, 0, 1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}        {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|       2                2                 2                                  |
|      p          + 8131p          + 11912p          + 4574p         p        |
|       {0, 1, 0}        {0, 1, 1}         {1, 0, 0}        {0, 1, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                            2                                |
|         + 1159p         p          - 14795p          + 5387p         p      |
|      1}        {1, 0, 0} {1, 0, 1}         {1, 0, 1}        {0, 1, 0} {1, 1,|
|      -----------------------------------------------------------------------|
|                                            2                                |
|         - 4486p         p          + 15671p          - 3277p         p      |
|      0}        {1, 0, 0} {1, 1, 0}         {1, 1, 0}        {0, 1, 0} {1, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|         - 9205p         p          + 8997p         p          - 15475p      |
|      1}        {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}         {1, 0,|
|      -----------------------------------------------------------------------|
|                                                       2                     |
|        p          + 15686p         p          + 12435p         )            |
|      1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}         {1, 1, 1}             |
|                                                                             |
|o26 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i27 : I2 = intersectWithCImodel(V, L)                                        |
|                                                                             |
|                                                                             |
|o27 = ideal (p          - 11083p          - 13237p          - 771p         , |
|              {0, 0, 1}         {0, 1, 1}         {1, 0, 1}       {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      p          - 11083p          - 13237p          - 771p         , p      |
|       {0, 0, 0}         {0, 1, 0}         {1, 0, 0}       {1, 1, 0}   {1, 0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - p         p         , p         p          - p          |
|      1} {1, 1, 0}    {1, 0, 0} {1, 1, 1}   {0, 1, 1} {1, 1, 0}    {0, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         , p         p          + 705p         p          + 9353p   |
|      0} {1, 1, 1}   {0, 1, 0} {1, 0, 1}       {0, 1, 1} {1, 0, 1}        {1,|
|      -----------------------------------------------------------------------|
|                             2                                               |
|           p          - 8959p          + 5062p         p          - 1880p    |
|      0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1, 0} {1, 1, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p          + 15763p         p          + 11084p         p         |
|      1, 1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}|
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      - 12581p         p          - 13990p         , p         p          +  |
|              {1, 1, 0} {1, 1, 1}         {1, 1, 1}   {0, 1, 1} {1, 0, 0}    |
|      -----------------------------------------------------------------------|
|                                                                2            |
|      705p         p          + 9353p         p          - 8959p          +  |
|          {0, 1, 1} {1, 0, 1}        {1, 0, 0} {1, 0, 1}        {1, 0, 1}    |
|      -----------------------------------------------------------------------|
|                                                                             |
|      5062p         p          - 1880p         p          + 15763p           |
|           {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}         {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 11084p         p          - 12581p         p          - |
|      0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}   |
|      -----------------------------------------------------------------------|
|            2                                      2                         |
|      13990p         , p         p          + 9353p          + 15023p        |
|            {1, 1, 1}   {0, 1, 0} {1, 0, 0}        {1, 0, 0}         {0, 1,  |
|      -----------------------------------------------------------------------|
|                                                       2                     |
|        p          - 10206p         p          + 11504p          + 5062p     |
|      1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}         {1, 0, 1}        {0,  |
|      -----------------------------------------------------------------------|
|                                                          2                  |
|           p          + 15763p         p          - 12581p          +        |
|      1, 0} {1, 1, 0}         {1, 0, 0} {1, 1, 0}         {1, 1, 0}          |
|      -----------------------------------------------------------------------|
|                                                                             |
|      13746p         p          + 13277p         p          + 3210p          |
|            {0, 1, 0} {1, 1, 1}         {0, 1, 1} {1, 1, 1}        {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 5488p         p          - 9216p         p          +   |
|      0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}     |
|      -----------------------------------------------------------------------|
|           2                                      2                          |
|      6026p         , p         p          - 7547p          + 13612p         |
|           {1, 1, 1}   {0, 1, 0} {0, 1, 1}        {0, 1, 1}         {0, 1,   |
|      -----------------------------------------------------------------------|
|                                                      2                      |
|        p          + 11912p         p          + 2824p          + 5387p      |
|      1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 13671p         p          - 4486p         p          -  |
|      0} {1, 1, 1}         {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                                                                  2          |
|      8494p         p          + 15671p         p          - 6785p         , |
|           {1, 0, 1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}        {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|       2                2                 2                                  |
|      p          + 8131p          + 11912p          + 4574p         p        |
|       {0, 1, 0}        {0, 1, 1}         {1, 0, 0}        {0, 1, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                            2                                |
|         + 1159p         p          - 14795p          + 5387p         p      |
|      1}        {1, 0, 0} {1, 0, 1}         {1, 0, 1}        {0, 1, 0} {1, 1,|
|      -----------------------------------------------------------------------|
|                                            2                                |
|         - 4486p         p          + 15671p          - 3277p         p      |
|      0}        {1, 0, 0} {1, 1, 0}         {1, 1, 0}        {0, 1, 0} {1, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|         - 9205p         p          + 8997p         p          - 15475p      |
|      1}        {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}         {1, 0,|
|      -----------------------------------------------------------------------|
|                                                       2                     |
|        p          + 15686p         p          + 12435p         )            |
|      1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}         {1, 1, 1}             |
|                                                                             |
|o27 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i28 : I1 == I2                                                               |
|                                                                             |
|o28 = true                                                                   |
+-----------------------------------------------------------------------------+

The Verbose=>true option prints the progress of each step in the saturation
process - a message is printed after saturating the ideal $V$, the conditional
independence ideal $I$, and the sum $V + I$ with respect to each hyperplane of
the probability simplex.

+-----------------------------------------------------+
|i29 : FF = ZZ/32003;                                 |
+-----------------------------------------------------+
|i30 : d = {2,3,2};                                   |
+-----------------------------------------------------+
|i31 : X = randomGame(d, CoefficientRing => FF);      |
+-----------------------------------------------------+
|i32 : PR = probabilityRing(d, CoefficientRing => FF);|
+-----------------------------------------------------+
|i33 : V = spohnIdeal(PR, X);                         |
|                                                     |
|o33 : Ideal of PR                                    |
+-----------------------------------------------------+
|i34 : L = {{{1,2},{3},{}}};                          |
+-----------------------------------------------------+
|i35 : I = intersectWithCImodel(V, L, Verbose=>true); |
|Completed step 1 of saturating CI ideal              |
|Completed step 1 of saturating input ideal           |
|Completed step 2 of saturating CI ideal              |
|Completed step 2 of saturating input ideal           |
|Completed step 3 of saturating CI ideal              |
|Completed step 3 of saturating input ideal           |
|Completed step 4 of saturating CI ideal              |
|Completed step 4 of saturating input ideal           |
|Completed step 5 of saturating CI ideal              |
|Completed step 5 of saturating input ideal           |
|Completed step 6 of saturating CI ideal              |
|Completed step 6 of saturating input ideal           |
|Completed step 7 of saturating CI ideal              |
|Completed step 7 of saturating input ideal           |
|Completed step 8 of saturating CI ideal              |
|Completed step 8 of saturating input ideal           |
|Completed step 9 of saturating CI ideal              |
|Completed step 9 of saturating input ideal           |
|Completed step 10 of saturating CI ideal             |
|Completed step 10 of saturating input ideal          |
|Completed step 11 of saturating CI ideal             |
|Completed step 11 of saturating input ideal          |
|Completed step 12 of saturating CI ideal             |
|Completed step 12 of saturating input ideal          |
|Completed step 13 of saturating CI ideal             |
|Completed step 13 of saturating input ideal          |
|Completed step 1 of saturating sum                   |
|Completed step 2 of saturating sum                   |
|Completed step 3 of saturating sum                   |
|Completed step 4 of saturating sum                   |
|Completed step 5 of saturating sum                   |
|Completed step 6 of saturating sum                   |
|Completed step 7 of saturating sum                   |
|Completed step 8 of saturating sum                   |
|Completed step 9 of saturating sum                   |
|Completed step 10 of saturating sum                  |
|Completed step 11 of saturating sum                  |
|Completed step 12 of saturating sum                  |
|                                                     |
|o35 : Ideal of PR                                    |
+-----------------------------------------------------+

See also
========

  * conditionalIndependenceIdeal (GraphicalModels)
    -- ideal of a list of
    conditional independent statements
  * mapToProbabilityRing -- ring isomorphism to
    the given probabilityRing from the corresponding markovRing
  * toMarkovRing -- ring of joint probability
    distributions created with the markovRing function from the GraphicalModels
    package
  * ciIdeal -- the ideal of a list of conditional independence
    statements
  * globalMarkov (GraphicalModels) -- global Markov
    statements for a graph or a directed graph

Ways to use intersectWithCImodel:
=================================

  * "intersectWithCImodel(Ideal,Graph)"
  * "intersectWithCImodel(Ideal,Graph,List)"
  * "intersectWithCImodel(Ideal,List)"
  * "intersectWithCImodel(Ideal,List,List)"

For the programmer
==================

The object intersectWithCImodel, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:2061:0.

konstanzMatrix -- construct the Konstanz matrix of a given game
***************************************************************

  * Usage:
        konstanzMatrix(PR, X)
  * Inputs:
      * PR, a ring (Macaulay2Doc), a probability ring obtained via
        probabilityRing(Di), where $Di = \{d_1 , \cdots , d_n\}$ is the format
        of the game
      * X, a list (Macaulay2Doc), a list of n tensors of format
        $D_i$ describing the payoffs of the game
  * Optional inputs (Macaulay2Doc),:
      * KonstanzVariableName (missing documentation) => ..., default value "k",
  * Outputs:
      * a matrix (Macaulay2Doc), the $(d_1 + \ldots + d_n) \times
        (d-1 \cdots d_n)$-dimensional Konstanz matrix

Description
===========

It is crucial that the formats in PR and X match up. The Konstanz matrix
$K_X(k)$ is the unique matrix with monic polynomials as entries such that the
Spohn variety is the union $\bigcup_{k \in (\mathbb P^1)^n} \ker K_X(k)$.

+-----------------------------------------------------------------------------+
|i1 : Di = {2,2,3};                                                           |
+-----------------------------------------------------------------------------+
|i2 : PR = probabilityRing(Di);                                               |
+-----------------------------------------------------------------------------+
|i3 : X = randomGame(Di);                                                     |
+-----------------------------------------------------------------------------+
|i4 : K = konstanzMatrix(PR,X)                                                |
|                                                                             |
|o4 = | k_0-9/2 k_0-9/4 k_0-3/4  k_0-7/4 k_0-7/9 k_0-7/10 0        0          |
|     | 0       0       0        0       0       0        k_0-7/10 k_0-7/3    |
|     | k_1-5/4 k_1-2/9 k_1-3/10 0       0       0        k_1-10   k_1-3/2    |
|     | 0       0       0        k_1-3/7 k_1-5   k_1-10/9 0        0          |
|     | k_2-5/3 0       0        k_2-6/5 0       0        k_2-5/3  0          |
|     | 0       k_2-7/2 0        0       k_2-5/7 0        0        k_2-1/10   |
|     | 0       0       k_2-2/5  0       0       k_2-5/9  0        0          |
|     ------------------------------------------------------------------------|
|     0       0       0        0       |                                      |
|     k_0-7   k_0-3/7 k_0-6/7  k_0-6   |                                      |
|     k_1-7/8 0       0        0       |                                      |
|     0       k_1-5/6 k_1-5    k_1-2/5 |                                      |
|     0       k_2-3/7 0        0       |                                      |
|     0       0       k_2-9/10 0       |                                      |
|     k_2-4/3 0       0        k_2-4/7 |                                      |
|                                                                             |
|                        7                 12                                 |
|o4 : Matrix (PR[k ..k ])  <-- (PR[k ..k ])                                   |
|                 0   2             0   2                                     |
+-----------------------------------------------------------------------------+

The optional argument KonstanzVariableName allows to change the name of the
variables. If no variable name choice is specified, the variables will be named
with k.

+----------------------------------------------------------+
|i5 : Di = {2,2};                                          |
+----------------------------------------------------------+
|i6 : PR = probabilityRing(Di);                            |
+----------------------------------------------------------+
|i7 : X = randomGame(Di);                                  |
+----------------------------------------------------------+
|i8 : K = konstanzMatrix(PR,X, KonstanzVariableName => "z")|
|                                                          |
|o8 = | z_0-5/9 z_0-5/9 0       0       |                  |
|     | 0       0       z_0-6/7 z_0-6   |                  |
|     | z_1-5/4 0       z_1-8/5 0       |                  |
|     | 0       z_1-2/9 0       z_1-9/4 |                  |
|                                                          |
|                        4                 4               |
|o8 : Matrix (PR[z ..z ])  <-- (PR[z ..z ])                |
|                 0   1             0   1                  |
+----------------------------------------------------------+

See also
========

  * probabilityRing -- ring of probability
    distributions of a game indexed by ordered multi-indices
  * randomGame -- construct a game of a given format with
    arbitrary payoffs
  * spohnMatrices -- compute the list of Spohn matrices
    of a given game
  * spohnIdeal -- compute the ideal of the Spohn variety of
    a given game

Ways to use konstanzMatrix:
===========================

  * konstanzMatrix(Ring,List) (missing documentation)

For the programmer
==================

The object konstanzMatrix, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1740:0.

KonstanzVariableName
********************

Functions with optional argument named KonstanzVariableName:
============================================================

  * konstanzMatrix(...,KonstanzVariableName=>...) (missing documentation)

For the programmer
==================

The object KonstanzVariableName (missing documentation) is a symbol (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
/home/runner/work/M2/M2/M2/BUILD/build/Macaulay2/packages/currentString:2:0.

mapToMarkovRing -- ring isomorphism from the given probabilityRing to the corresponding markovRing
**************************************************************************************************

  * Usage:
        mapToMarkovRing R
  * Inputs:
      * R, a ring (Macaulay2Doc), must be a probabilityRing
  * Outputs:
      * a ring map (Macaulay2Doc), the isomorphism identifying R
        with toMarkovRing(R). The variable $p_{\{i_1, \ldots, i_k\}}$ is sent
        to $q_{(i_1+1, \dots , i_k+1)}$.

Description
===========

This function creates the RingMap from a given probabilityRing to its
canonically isomorphic markovRing.

+-----------------------------------------------------------------------------+
|i1 : R = probabilityRing {2,3,4};                                            |
+-----------------------------------------------------------------------------+
|i2 : markovR = toMarkovRing R;                                               |
+-----------------------------------------------------------------------------+
|i3 : F = mapToMarkovRing R                                                   |
|                                                                             |
|o3 = map (markovR, R, {q     , q     , q     , q     , q     , q     ,       |
|                        1,1,1   1,1,2   1,1,3   1,1,4   1,2,1   1,2,2        |
|                                                                             |
|o3 : RingMap markovR <-- R                                                   |
|-----------------------------------------------------------------------------|
|q     , q     , q     , q     , q     , q     , q     , q     , q     ,      |
| 1,2,3   1,2,4   1,3,1   1,3,2   1,3,3   1,3,4   2,1,1   2,1,2   2,1,3       |
|-----------------------------------------------------------------------------|
|q     , q     , q     , q     , q     , q     , q     , q     , q     })     |
| 2,1,4   2,2,1   2,2,2   2,2,3   2,2,4   2,3,1   2,3,2   2,3,3   2,3,4       |
+-----------------------------------------------------------------------------+
|i4 : target F                                                                |
|                                                                             |
|o4 = markovR                                                                 |
|                                                                             |
|o4 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i5 : source F                                                                |
|                                                                             |
|o5 = R                                                                       |
|                                                                             |
|o5 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i6 : isInjective F                                                           |
|                                                                             |
|o6 = true                                                                    |
+-----------------------------------------------------------------------------+
|i7 : F.matrix                                                                |
|                                                                             |
|o7 = | q_(1,1,1) q_(1,1,2) q_(1,1,3) q_(1,1,4) q_(1,2,1) q_(1,2,2) q_(1,2,3) |
|     ------------------------------------------------------------------------|
|     q_(1,2,4) q_(1,3,1) q_(1,3,2) q_(1,3,3) q_(1,3,4) q_(2,1,1) q_(2,1,2)   |
|     ------------------------------------------------------------------------|
|     q_(2,1,3) q_(2,1,4) q_(2,2,1) q_(2,2,2) q_(2,2,3) q_(2,2,4) q_(2,3,1)   |
|     ------------------------------------------------------------------------|
|     q_(2,3,2) q_(2,3,3) q_(2,3,4) |                                         |
|                                                                             |
|                   1            24                                           |
|o7 : Matrix markovR  <-- markovR                                             |
+-----------------------------------------------------------------------------+

See also
========

  * toMarkovRing -- ring of joint probability
    distributions created with the markovRing function from the GraphicalModels
    package
  * mapToProbabilityRing -- ring isomorphism to
    the given probabilityRing from the corresponding markovRing

Ways to use mapToMarkovRing:
============================

  * "mapToMarkovRing(Ring)"

For the programmer
==================

The object mapToMarkovRing, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1817:0.

mapToProbabilityRing -- ring isomorphism to the given probabilityRing from the corresponding markovRing
*******************************************************************************************************

  * Usage:
        mapToProbabilityRing R
  * Inputs:
      * R, a ring (Macaulay2Doc), must be a probabilityRing
  * Outputs:
      * a ring map (Macaulay2Doc), the isomorphism identifying R
        with toMarkovRing(R). The variable $q_{(i_1+1, \dots , i_k+1)}$ is sent
        to $p_{\{i_1, \ldots, i_k\}}$.

Description
===========

This function creates the RingMap to a given probabilityRing from its
canonically isomorphic markovRing.

+-----------------------------------------------------------------------------+
|i1 : R = probabilityRing {2,3,4};                                            |
+-----------------------------------------------------------------------------+
|i2 : markovR = toMarkovRing R;                                               |
+-----------------------------------------------------------------------------+
|i3 : F = mapToProbabilityRing R                                              |
|                                                                             |
|o3 = map (R, markovR, {p         , p         , p         , p         , p     |
|                        {0, 0, 0}   {0, 0, 1}   {0, 0, 2}   {0, 0, 3}   {0,  |
|                                                                             |
|o3 : RingMap R <-- markovR                                                   |
|-----------------------------------------------------------------------------|
|     , p         , p         , p         , p         , p         , p         |
|1, 0}   {0, 1, 1}   {0, 1, 2}   {0, 1, 3}   {0, 2, 0}   {0, 2, 1}   {0, 2,   |
|-----------------------------------------------------------------------------|
|  , p         , p         , p         , p         , p         , p         ,  |
|2}   {0, 2, 3}   {1, 0, 0}   {1, 0, 1}   {1, 0, 2}   {1, 0, 3}   {1, 1, 0}   |
|-----------------------------------------------------------------------------|
|p         , p         , p         , p         , p         , p         , p    |
| {1, 1, 1}   {1, 1, 2}   {1, 1, 3}   {1, 2, 0}   {1, 2, 1}   {1, 2, 2}   {1, |
|-----------------------------------------------------------------------------|
|     })                                                                      |
|2, 3}                                                                        |
+-----------------------------------------------------------------------------+
|i4 : target F                                                                |
|                                                                             |
|o4 = R                                                                       |
|                                                                             |
|o4 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i5 : source F                                                                |
|                                                                             |
|o5 = markovR                                                                 |
|                                                                             |
|o5 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i6 : isInjective F                                                           |
|                                                                             |
|o6 = true                                                                    |
+-----------------------------------------------------------------------------+
|i7 : F.matrix                                                                |
|                                                                             |
|o7 = | p_{0, 0, 0} p_{0, 0, 1} p_{0, 0, 2} p_{0, 0, 3} p_{0, 1, 0} p_{0, 1,  |
|     ------------------------------------------------------------------------|
|     1} p_{0, 1, 2} p_{0, 1, 3} p_{0, 2, 0} p_{0, 2, 1} p_{0, 2, 2} p_{0, 2, |
|     ------------------------------------------------------------------------|
|     3} p_{1, 0, 0} p_{1, 0, 1} p_{1, 0, 2} p_{1, 0, 3} p_{1, 1, 0} p_{1, 1, |
|     ------------------------------------------------------------------------|
|     1} p_{1, 1, 2} p_{1, 1, 3} p_{1, 2, 0} p_{1, 2, 1} p_{1, 2, 2} p_{1, 2, |
|     ------------------------------------------------------------------------|
|     3} |                                                                    |
|                                                                             |
|             1      24                                                       |
|o7 : Matrix R  <-- R                                                         |
+-----------------------------------------------------------------------------+

See also
========

  * toMarkovRing -- ring of joint probability
    distributions created with the markovRing function from the GraphicalModels
    package
  * mapToProbabilityRing -- ring isomorphism to
    the given probabilityRing from the corresponding markovRing

Ways to use mapToProbabilityRing:
=================================

  * "mapToProbabilityRing(Ring)"

For the programmer
==================

The object mapToProbabilityRing, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1855:0.

nashEquilibriumIdeal(Ring,List) -- make the Nash Equilibrium ideal
******************************************************************

  * Function: nashEquilibriumIdeal,
  * Usage:
        nashEquilibriumIdeal(R, L)
  * Inputs:
      * R, a ring (Macaulay2Doc), the Nash Equilibrium ring.
        Typically obtained via nashEquilibriumRing,
      * L, a list (Macaulay2Doc), a list of n payoff tensors
  * Outputs:
      * an ideal (Macaulay2Doc), an ideal in the Nash Equilibrium
        ring R generated by the Nash equilibrium polynomials, along with the
        linear relations of the probability variables

Description
===========

For an $n$-player game, the totally mixed Nash equilibria are the zero loci of
a system of polynomials in the interior of product of probability simplices.
The coefficients of these polynomials are certain differences of the entries of
the payoff tensors. These polynomials, together with the linear constraints
$\sum^{d_i -1}_{j=0} p_{i,j}=1$ for each player $i$, generate an ideal in the
polynomial ring computed by nashEquilibriumRing.

+-----------------------------------------------------------------------------+
|i1 : tensors = randomGame {2,2,2};                                           |
+-----------------------------------------------------------------------------+
|i2 : R = nashEquilibriumRing tensors;                                        |
+-----------------------------------------------------------------------------+
|i3 : I = nashEquilibriumIdeal(R, tensors)                                    |
|                                                                             |
|              67                  1                 31                  7    |
|o3 = ideal (- --p      p       - --p      p       - --p      p       + --p   |
|              18 {1, 0} {2, 0}   20 {1, 1} {2, 0}   20 {1, 0} {2, 1}   12 {1,|
|     ------------------------------------------------------------------------|
|                  43                 19                 39                   |
|       p      , - --p      p       - --p      p       + --p      p       +   |
|     1} {2, 1}     7 {0, 0} {2, 0}   20 {0, 1} {2, 0}    7 {0, 0} {2, 1}     |
|     ------------------------------------------------------------------------|
|     13                  35                  1                 17            |
|     --p      p      , - --p      p       - --p      p       - --p      p    |
|     63 {0, 1} {2, 1}     9 {0, 0} {1, 0}   24 {0, 1} {1, 0}    2 {0, 0} {1, |
|     ------------------------------------------------------------------------|
|          23                                                                 |
|        - --p      p      , p       + p       - 1, p       + p       - 1,    |
|     1}    5 {0, 1} {1, 1}   {0, 0}    {0, 1}       {1, 0}    {1, 1}         |
|     ------------------------------------------------------------------------|
|     p       + p       - 1)                                                  |
|      {2, 0}    {2, 1}                                                       |
|                                                                             |
|o3 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+

An introduction together with the relevant definitions is given in Chapter 6,
Sturmfels, Bernd, Solving Systems of Polynomial Equations. American
Mathematical Society, 2002 and in Abo, Hirotachi, Portakal, Irem, and Sodomaco,
Luca, A vector bundle approach to Nash equilibria, arXiv:2504.03456.

See also
========

  * nashEquilibriumRing -- define the
    Nash Equilibrium ring
  * deltaList -- generate the list of Newton
    polytopes for a generic game
  * blockDerangements -- compute the block
    derangements
  * numberTMNE -- compute the maximum number of
    totally mixed Nash equilibria for a generic game

Ways to use this method:
========================

  * nashEquilibriumIdeal(Ring,List)
    -- make the Nash Equilibrium ideal
-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1371:0.

nashEquilibriumRing(List) -- define the Nash Equilibrium ring
*************************************************************

  * Function: nashEquilibriumRing,
  * Usage:
        nashEquilibriumRing L
  * Inputs:
      * L, a list (Macaulay2Doc), a list of n-see Tensor,
        s with same dimensions
  * Outputs:
      * a ring (Macaulay2Doc), a polynomial ring generated by mixed
        strategies

Description
===========

Let $p_{i,j}$ be the probability of $i$-th player choosing the $j$-th strategy,
where $j \in \{0, \cdots, d_j-1\}$. The ideal of the totally mixed Nash
equilibria of the game is defined in the polynomial ring over a field $k$ with
generators $\{p_{i,j} \ | \ 0\leq i\leq n-1, 0\leq j\leq d_j-1\}$. The ring
generators are ordered lexicographically.

+-----------------------------------------------------------------------------+
|i1 : tensors = randomGame {2,4,3};                                           |
+-----------------------------------------------------------------------------+
|i2 : R = nashEquilibriumRing tensors;                                        |
+-----------------------------------------------------------------------------+
|i3 : baseRing R                                                              |
|                                                                             |
|o3 = QQ                                                                      |
|                                                                             |
|o3 : Ring                                                                    |
+-----------------------------------------------------------------------------+
|i4 : gens R                                                                  |
|                                                                             |
|o4 = {p      , p      , p      , p      , p      , p      , p      , p      ,|
|       {0, 0}   {0, 1}   {1, 0}   {1, 1}   {1, 2}   {1, 3}   {2, 0}   {2, 1} |
|     ------------------------------------------------------------------------|
|     p      }                                                                |
|      {2, 2}                                                                 |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * nashEquilibriumIdeal -- make
    the Nash Equilibrium ideal
  * deltaList -- generate the list of Newton
    polytopes for a generic game
  * blockDerangements -- compute the block
    derangements
  * numberTMNE -- compute the maximum number of
    totally mixed Nash equilibria for a generic game

Ways to use this method:
========================

  * nashEquilibriumRing(List) -- define
    the Nash Equilibrium ring
-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1332:0.

numberTMNE(List) -- compute the maximum number of totally mixed Nash equilibria for a generic game
**************************************************************************************************

  * Function: numberTMNE,
  * Usage:
        numberTMNE d
  * Inputs:
      * d, a list (Macaulay2Doc), a list of integers representing
        the format of the game
  * Outputs:
      * an integer (Macaulay2Doc), an integer value, the degree of
        the top Chern class of a vector bundle on a product of projective
        spaces, representing the maximum number of totally mixed Nash
        equilibria of a generic game of format $\mathbf{d}$.

Description
===========

For an $n$-player game where the $i$-th player has $d_i$ pure strategies, the
maximum number of isolated totally mixed Nash equilibria is given by the degree
$c(\mathbf{d})$ of the top Chern class of the following vector bundle:

\[ E \coloneqq \bigoplus_{i=0}^{n-1}
\ko_{\mathbb{P}^{\mathbf{d}}}({\mathbf{1}}_i)^{\oplus(d_i-1)},\]

where $\mathbb{P}^{\mathbf{d}}=\prod_{i=0}^{n-1}\mathbb{P}^{d_i-1}$ and
$\mathbf{1}_i=(1,\ldots,1,0,1,\ldots,1)$, where the entry $0$ is in the $i$-th
component. In particular, this function computes the integer $c(\mathbf{d})$ as
the coefficient of the monomial $\prod_{i=0}^{n-1} h_i^{d_i-1}$ in
$\prod_{i=0}^{n-1} \hat{h}_i^{d_i-1}$ with $\hat{h}_i\coloneqq \sum_{j\neq
i}h_j$, where $h_i$ denotes the pullback of the hyperplane class on the $i$-th
factor $\mathbb{P}^{d_i-1}$ of $\PP^\bd$ via the projection map.

+-------------------------+
|i1 : d = {2,2,2};        |
+-------------------------+
|i2 : nTMNE = numberTMNE d|
|                         |
|o2 = 2                   |
+-------------------------+

Alternatively, if you have a tensor $T$, you can compute its maximum number as
follows:

+---------------------------------+
|i3 : T = randomTensor {2,2,2}    |
|                                 |
|o3 = Tensor{...11...}            |
|                                 |
|o3 : Tensor                      |
+---------------------------------+
|i4 : nTMNE2 = numberTMNE format T|
|                                 |
|o4 = 2                           |
+---------------------------------+

See also
========

  * blockDerangements -- compute the block
    derangements

Ways to use this method:
========================

  * numberTMNE(List) -- compute the maximum number
    of totally mixed Nash equilibria for a generic game
-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1455:0.

probabilityRing -- ring of probability distributions of a game indexed by ordered multi-indices
***********************************************************************************************

  * Usage:
        probabilityRing(Di)
  * Inputs:
      * Di, a list (Macaulay2Doc), a list of natural numbers
        $d_0,\dots,d_{n-1}$
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing (missing documentation) => ..., default value QQ,
      * ProbabilityVariableName (missing documentation) => ..., default value
        "p",
  * Outputs:
      * a ring (Macaulay2Doc), a polynomial ring with a tensor of
        variables $p_{i_0,\dots,i_{n-1}}$ such that $i_j$ runs from $0$ to
        $d_j-1$.

Description
===========

The list $Di$ represents the format of the game. In this example, we create a
ring of probability distributions coming from a game of format {2, 3, 2}. This
format can be accessed from the ring through the field "gameFormat".

The variables $p#i$ are the entries of the tensor $p$, which can be accessed
from the ring through the field "probabilityVariable".

+-----------------------------------------------------------------------------+
|i1 : Di = {2,3,2};                                                           |
+-----------------------------------------------------------------------------+
|i2 : PR = probabilityRing Di;                                                |
+-----------------------------------------------------------------------------+
|i3 : numgens PR                                                              |
|                                                                             |
|o3 = 12                                                                      |
+-----------------------------------------------------------------------------+
|i4 : pairs PR#"probabilityVariable"                                          |
|                                                                             |
|o4 = {(indexes, {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {0, 2, 0}, {0,  |
|                                                                             |
|     ------------------------------------------------------------------------|
|     2, 1}, {1, 0, 0}, {1, 0, 1}, {1, 1, 0}, {1, 1, 1}, {1, 2, 0}, {1, 2,    |
|                                                                             |
|     ------------------------------------------------------------------------|
|     1}}), ({1, 1, 0}, p         ), ({1, 1, 1}, p         ), ({1, 0, 0}, p   |
|                        {1, 1, 0}                {1, 1, 1}                {1,|
|     ------------------------------------------------------------------------|
|          ), ({0, 2, 0}, p         ), ({1, 0, 1}, p         ), ({0, 2, 1},   |
|     0, 0}                {0, 2, 0}                {1, 0, 1}                 |
|     ------------------------------------------------------------------------|
|     p         ), ({0, 1, 0}, p         ), ({0, 1, 1}, p         ), (format, |
|      {0, 2, 1}                {0, 1, 0}                {0, 1, 1}            |
|     ------------------------------------------------------------------------|
|     {2, 3, 2}), ({0, 0, 0}, p         ), ({0, 0, 1}, p         ), ({1, 2,   |
|                              {0, 0, 0}                {0, 0, 1}             |
|     ------------------------------------------------------------------------|
|     0}, p         ), ({1, 2, 1}, p         ), (coefficients, PR)}           |
|          {1, 2, 0}                {1, 2, 1}                                 |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+

The optional argument "CoefficientRing" allows to change the base field. If no
choice is specified, the base field is set to QQ. It is also possible to change
the name of the variable tensor through the optional argument
"ProbabilityVariableName", which is set to the string "p" by default.

+-----------------------------------------------------------------------------+
|i5 : PR2 = probabilityRing (Di, CoefficientRing=>RR,                         |
|-----------------------------------------------------------------------------|
|ProbabilityVariableName=>"q");                                               |
+-----------------------------------------------------------------------------+
|i6 : coefficientRing PR2                                                     |
|                                                                             |
|o6 = RR                                                                      |
|       53                                                                    |
|                                                                             |
|o6 : RealField                                                               |
+-----------------------------------------------------------------------------+
|i7 : pairs PR2#"probabilityVariable"                                         |
|                                                                             |
|o7 = {(indexes, {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {0, 2, 0}, {0,  |
|                                                                             |
|     ------------------------------------------------------------------------|
|     2, 1}, {1, 0, 0}, {1, 0, 1}, {1, 1, 0}, {1, 1, 1}, {1, 2, 0}, {1, 2,    |
|                                                                             |
|     ------------------------------------------------------------------------|
|     1}}), ({1, 1, 0}, q         ), ({1, 1, 1}, q         ), ({1, 0, 0}, q   |
|                        {1, 1, 0}                {1, 1, 1}                {1,|
|     ------------------------------------------------------------------------|
|          ), ({0, 2, 0}, q         ), ({1, 0, 1}, q         ), ({0, 2, 1},   |
|     0, 0}                {0, 2, 0}                {1, 0, 1}                 |
|     ------------------------------------------------------------------------|
|     q         ), ({0, 1, 0}, q         ), ({0, 1, 1}, q         ), (format, |
|      {0, 2, 1}                {0, 1, 0}                {0, 1, 1}            |
|     ------------------------------------------------------------------------|
|     {2, 3, 2}), ({0, 0, 0}, q         ), ({0, 0, 1}, q         ), ({1, 2,   |
|                              {0, 0, 0}                {0, 0, 1}             |
|     ------------------------------------------------------------------------|
|     0}, q         ), ({1, 2, 1}, q         ), (coefficients, PR2)}          |
|          {1, 2, 0}                {1, 2, 1}                                 |
|                                                                             |
|o7 : List                                                                    |
+-----------------------------------------------------------------------------+

Some functions such as spohnIdeal, konstanzMatrix,
see ciIdeal, or spohnCI,  require
the ring to be created by this function or in a similar manner.

See also
========

  * spohnIdeal -- compute the ideal of the Spohn variety of
    a given game
  * konstanzMatrix -- construct the Konstanz matrix of a
    given game
  * ciIdeal -- the ideal of a list of conditional independence
    statements
  * spohnCI -- ideal of the Spohn conditional independence (CI)
    variety

Ways to use probabilityRing:
============================

  * "probabilityRing(List)"

For the programmer
==================

The object probabilityRing, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1546:0.

probabilitySumIdeal -- ideal enforcing that a probability distribution sums to 1
********************************************************************************

  * Usage:
        probabilitySumIdeal R
  * Inputs:
      * R, a ring (Macaulay2Doc), a ring typically from
        probabilityRing,
  * Outputs:
      * an ideal (Macaulay2Doc), the ideal generated by the sum of
        all probabilities minus 1

Description
===========

Constructs the ideal expressing normalization of a joint probability
distribution. If given a list, it first calls probabilityRing,
with default options.

+-----------------------------------------------------------------------------+
|i1 : R = probabilityRing {2,3,4};                                            |
+-----------------------------------------------------------------------------+
|i2 : probabilitySumIdeal R                                                   |
|                                                                             |
|o2 = ideal(p          + p          + p          + p          + p          +  |
|            {0, 0, 0}    {0, 0, 1}    {0, 0, 2}    {0, 0, 3}    {0, 1, 0}    |
|     ------------------------------------------------------------------------|
|     p          + p          + p          + p          + p          + p      |
|      {0, 1, 1}    {0, 1, 2}    {0, 1, 3}    {0, 2, 0}    {0, 2, 1}    {0, 2,|
|     ------------------------------------------------------------------------|
|        + p          + p          + p          + p          + p          +   |
|     2}    {0, 2, 3}    {1, 0, 0}    {1, 0, 1}    {1, 0, 2}    {1, 0, 3}     |
|     ------------------------------------------------------------------------|
|     p          + p          + p          + p          + p          + p      |
|      {1, 1, 0}    {1, 1, 1}    {1, 1, 2}    {1, 1, 3}    {1, 2, 0}    {1, 2,|
|     ------------------------------------------------------------------------|
|        + p          + p          - 1)                                       |
|     1}    {1, 2, 2}    {1, 2, 3}                                            |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+

See also
========

  * probabilityRing -- ring of probability
    distributions of a game indexed by ordered multi-indices
  * spohnIdeal -- compute the ideal of the Spohn variety of
    a given game
  * spohnCI -- ideal of the Spohn conditional independence (CI)
    variety

Ways to use probabilitySumIdeal:
================================

  * "probabilitySumIdeal(Ring)"

For the programmer
==================

The object probabilitySumIdeal, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1577:0.

ProbabilityVariableName
***********************

Functions with optional argument named ProbabilityVariableName:
===============================================================

  * probabilityRing(...,ProbabilityVariableName=>...) (missing documentation)

For the programmer
==================

The object ProbabilityVariableName (missing documentation) is a symbol (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in
/home/runner/work/M2/M2/M2/BUILD/build/Macaulay2/packages/currentString:2:0.

randomGame -- construct a game of a given format with arbitrary payoffs
***********************************************************************

  * Usage:
        randomGame(Di)
  * Inputs:
      * Di, a list (Macaulay2Doc), a list describing the format of
        the game
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing (missing documentation) => ..., default value QQ,
  * Outputs:
      * a list (Macaulay2Doc), a list of n tensors of the given
        format that are the payoff tensors of a random game

Description
===========

The list $Di$ represents the format of the game. This example creates a random
game of format $2 \times 2$.

+--------------------------------------------------------+
|i1 : X = randomGame({2,2});                             |
+--------------------------------------------------------+
|i2 : peek X#0                                           |
|                                                        |
|o2 = Tensor{coefficients => QQ                         }|
|            format => {2, 2}                            |
|            indexes => {{0, 0}, {0, 1}, {1, 0}, {1, 1}} |
|                      3                                 |
|            {1, 0} => -                                 |
|                      4                                 |
|                      7                                 |
|            {1, 1} => -                                 |
|                      4                                 |
|                      9                                 |
|            {0, 0} => -                                 |
|                      2                                 |
|                      9                                 |
|            {0, 1} => -                                 |
|                      4                                 |
+--------------------------------------------------------+
|i3 : peek X#1                                           |
|                                                        |
|o3 = Tensor{coefficients => QQ                         }|
|            format => {2, 2}                            |
|            indexes => {{0, 0}, {0, 1}, {1, 0}, {1, 1}} |
|                       7                                |
|            {0, 1} => --                                |
|                      10                                |
|                       7                                |
|            {1, 0} => --                                |
|                      10                                |
|                      7                                 |
|            {0, 0} => -                                 |
|                      9                                 |
|                      7                                 |
|            {1, 1} => -                                 |
|                      3                                 |
+--------------------------------------------------------+

The optional argument CoefficientRing allows to change the ring of payoffs. If
no coefficient choice is specified, the payoffs will be rational numbers. This
example creates a random game of format $2 \times 2$ with integer coefficients.

+--------------------------------------------------------+
|i4 : X = randomGame({2,2}, CoefficientRing => ZZ);      |
+--------------------------------------------------------+
|i5 : peek X#0                                           |
|                                                        |
|o5 = Tensor{{0, 0} => 6                                }|
|            {0, 1} => 0                                 |
|            {1, 0} => 2                                 |
|            {1, 1} => 6                                 |
|            coefficients => ZZ                          |
|            format => {2, 2}                            |
|            indexes => {{0, 0}, {0, 1}, {1, 0}, {1, 1}} |
+--------------------------------------------------------+
|i6 : peek X#1                                           |
|                                                        |
|o6 = Tensor{{0, 0} => 9                                }|
|            {0, 1} => 3                                 |
|            {1, 0} => 5                                 |
|            {1, 1} => 6                                 |
|            coefficients => ZZ                          |
|            format => {2, 2}                            |
|            indexes => {{0, 0}, {0, 1}, {1, 0}, {1, 1}} |
+--------------------------------------------------------+

Outputs of this function can be used as input for the functions
nashEquilibriumIdeal,
spohnMatrices, spohnIdeal, and
konstanzMatrix.

See also
========

  * nashEquilibriumIdeal -- make
    the Nash Equilibrium ideal
  * spohnMatrices -- compute the list of Spohn matrices
    of a given game
  * spohnIdeal -- compute the ideal of the Spohn variety of
    a given game
  * konstanzMatrix -- construct the Konstanz matrix of a
    given game

Ways to use randomGame:
=======================

  * randomGame(List) (missing documentation)

For the programmer
==================

The object randomGame, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1620:0.

randomTensor -- construct a tensor with random entries from a given ring
************************************************************************

  * Usage:
        randomTensor format
        randomTensor(R, format)
  * Inputs:
      * format, a list (Macaulay2Doc), A list of integers
        specifying the format of the tensor (e.g. {2,2,2} creates a 2x2x2
        tensor)
      * R, a ring (Macaulay2Doc), (Optional) A ring from which
        random coefficients will be drawn.
  * Outputs:
      * an instance of the type Tensor, A tensor whose entries
        are randomly selected elements of the ring.

Description
===========

This method constructs a tensor with the specified format and fills it with
random elements from the given ring. Internally, it uses a hash table where
each key is a multi-index (a list of positions) and the value is a random
element from the ring. Metadata such as the format, coefficient ring, and index
set are stored in the tensor as well.

+--------------------------------------------------------+
|i1 : T1 = randomTensor {2,2,2};                         |
+--------------------------------------------------------+
|i2 : T1#{0,1,1}                                         |
|                                                        |
|     7                                                  |
|o2 = -                                                  |
|     4                                                  |
|                                                        |
|o2 : QQ                                                 |
+--------------------------------------------------------+
|i3 : T2 = randomTensor(ZZ/101, {2,2});                  |
+--------------------------------------------------------+
|i4 : peek T2                                            |
|                                                        |
|o4 = Tensor{{0, 0} => 16                               }|
|            {0, 1} => 22                                |
|            {1, 0} => 45                                |
|            {1, 1} => -34                               |
|            format => {2, 2}                            |
|            indexes => {{0, 0}, {0, 1}, {1, 0}, {1, 1}} |
|                             ZZ                         |
|            coefficients => ---                         |
|                            101                         |
+--------------------------------------------------------+

See also
========

  * zeroTensor -- construct a tensor with zero entries from
    a given ring.

Ways to use randomTensor:
=========================

  * "randomTensor(List)"
  * "randomTensor(Ring,List)"

For the programmer
==================

The object randomTensor, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1107:0.

slice -- extract a slice of a tensor
************************************

  * Usage:
        slice(T, Lstart, Lend)
  * Inputs:
      * T, an instance of the type Tensor, A tensor from which
        to extract the slice.
      * Lstart, a list (Macaulay2Doc), A list of fixed indices
        before the varying position.
      * Lend, a list (Macaulay2Doc), A list of fixed indices after
        the varying position.
  * Outputs:
      * a list (Macaulay2Doc), A list of entries of the tensor
        along the specified dimension.

Description
===========

This method varies the index at the position given by the length of `Lstart`,
iterating from 0 to $d_i - 1$, where $d_i$ is the corresponding index of the
format.

The slice is formed by fixing all other indices and varying only the one at the
slicing position.

+---------------------------------+
|i1 : T = zeroTensor(QQ, {2,3,2});|
+---------------------------------+
|i2 : T#{0,0,0} = 5;              |
+---------------------------------+
|i3 : T#{0,1,0} = 6;              |
+---------------------------------+
|i4 : T#{0,2,0} = 7;              |
+---------------------------------+
|i5 : slice(T, {0}, {0})          |
|                                 |
|o5 = {5, 6, 7}                   |
|                                 |
|o5 : List                        |
+---------------------------------+

See also
========

  * enumerateTensorIndices -- generate index
    tuples for a tensor with given dimensions
  * Tensor -- a mutable hash table representing a tensor

Ways to use slice:
==================

  * "slice(Tensor,List,List)"

For the programmer
==================

The object slice, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1146:0.

spohnCI -- ideal of the Spohn conditional independence (CI) variety
*******************************************************************

  * Usage:
        spohnCI(PR, X, G)
        spohnCI(PR, X, G, PlayerNames)
        spohnCI(PR, X, Stmts)
        spohnCI(PR, X, Stmts, PlayerNames)
  * Inputs:
      * PR, a ring (Macaulay2Doc), the probability ring (must be
        created with probabilityRing)
      * X, a list (Macaulay2Doc), the n tensors defining the game
      * G, a graph (Graphs), the graph specifying the conditional
        independence conditions
      * Stmts, a list (Macaulay2Doc), a list of lists {L1,L2,L3}
        corresponding to the relation "L1 and L2 are conditionally independent
        given L3".
      * PlayerNames, a list (Macaulay2Doc), the ordered list of
        players - the names of the random variables in the conditional
        independence statements or vertices of the graph. If PlayerNames is
        omitted, the players (or the vertices of G) are assumed to be labelled
        1..n.
  * Optional inputs (Macaulay2Doc),:
      * Verbose (missing documentation) => ..., default value false,
  * Outputs:
      * an ideal (Macaulay2Doc), The ideal of the Spohn CI variety

Description
===========

spohnCI computes the ideal of the Spohn conditional independence variety for a
game $X$ and conditional independence model determined by an undirected graph
$G$ or set of conditional independence statements $Stmts$.

The input for the conditional independence model can be a set of conditional
probability statements or an undirected graph. A single conditional
independence statement is a list consisting of three disjoint lists of indices
for players, e.g. $\{ \{1,2\},\{4\}, \{3\} \}$ which represents the conditional
independence statement ``The strategies of Players 1 and 2 are conditionally
independent of Player 4's strategy given Player 3's strategy''.

Given an undirected graph $G$, the conditional independence statements are
produced via the globalMarkov function from the GraphicalModels package. A
global Markov statement for $G$ is a list $\{A, B, C\}$ of three disjoint lists
of vertices of $G$, where the subset $C$ separates the subset $A$ from the
subset $B$ in the graph $G$.

+-----------------------------------------------------------------------------+
|i1 : FF = ZZ/32003                                                           |
|                                                                             |
|o1 = FF                                                                      |
|                                                                             |
|o1 : QuotientRing                                                            |
+-----------------------------------------------------------------------------+
|i2 : d = {2,2,2};                                                            |
+-----------------------------------------------------------------------------+
|i3 : X = randomGame(d, CoefficientRing => FF);                               |
+-----------------------------------------------------------------------------+
|i4 : PR = probabilityRing(d, CoefficientRing => FF);                         |
+-----------------------------------------------------------------------------+
|i5 : G1 = graph ({}, Singletons => {1,2,3});                                 |
+-----------------------------------------------------------------------------+
|i6 : G2 = graph ({{1,2}}, Singletons => {3});                                |
+-----------------------------------------------------------------------------+
|i7 : I1 = spohnCI(PR,X,G1)                                                   |
|                                                                             |
|                                                                             |
|o7 = ideal (p          + 3034p          - 14745p         , p          +      |
|             {1, 0, 1}        {1, 1, 0}         {1, 1, 1}   {1, 0, 0}        |
|     ------------------------------------------------------------------------|
|                                                                             |
|     735p          - 10307p         , p          - 9548p          - 11660p   |
|         {1, 1, 0}         {1, 1, 1}   {0, 1, 1}        {1, 1, 0}         {1,|
|     ------------------------------------------------------------------------|
|                                                                             |
|          , p          + 6605p          - 4419p         , p          -       |
|     1, 1}   {0, 1, 0}        {1, 1, 0}        {1, 1, 1}   {0, 0, 1}         |
|     ------------------------------------------------------------------------|
|                                                                             |
|     9755p          - 8595p         , p          + 7770p          + 7307p    |
|          {1, 1, 0}        {1, 1, 1}   {0, 0, 0}        {1, 1, 0}        {1, |
|     ------------------------------------------------------------------------|
|             2                                            2                  |
|          , p          + 15142p         p          - 9838p         )         |
|     1, 1}   {1, 1, 0}         {1, 1, 0} {1, 1, 1}        {1, 1, 1}          |
|                                                                             |
|o7 : Ideal of PR                                                             |
+-----------------------------------------------------------------------------+
|i8 : I2 = spohnCI(PR,X,G2)                                                   |
|                                                                             |
|                                                                             |
|o8 = ideal (p          + 4385p          + 10227p          + 4120p         ,  |
|             {0, 0, 1}        {0, 1, 1}         {1, 0, 1}        {1, 1, 1}   |
|     ------------------------------------------------------------------------|
|                                                                             |
|     p          + 4385p          + 10227p          + 4120p         , p       |
|      {0, 0, 0}        {0, 1, 0}         {1, 0, 0}        {1, 1, 0}   {1, 0, |
|     ------------------------------------------------------------------------|
|                                                                             |
|       p          - p         p         , p         p          - p           |
|     1} {1, 1, 0}    {1, 0, 0} {1, 1, 1}   {0, 1, 1} {1, 1, 0}    {0, 1,     |
|     ------------------------------------------------------------------------|
|                                                                             |
|       p         , p         p          + 848p         p          + 12103p   |
|     0} {1, 1, 1}   {0, 1, 0} {1, 0, 1}       {0, 1, 1} {1, 0, 1}         {1,|
|     ------------------------------------------------------------------------|
|                            2                                                |
|          p          - 6085p          + 7894p         p          - 3300p     |
|     0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1, 0} {1, 1, 1}        {0,  |
|     ------------------------------------------------------------------------|
|                                                                             |
|          p          - 850p         p          - 10956p         p          + |
|     1, 1} {1, 1, 1}       {1, 0, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}   |
|     ------------------------------------------------------------------------|
|                                     2                                       |
|     6865p         p          + 8470p         , p         p          +       |
|          {1, 1, 0} {1, 1, 1}        {1, 1, 1}   {0, 1, 1} {1, 0, 0}         |
|     ------------------------------------------------------------------------|
|                                                                2            |
|     848p         p          + 12103p         p          - 6085p          +  |
|         {0, 1, 1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}        {1, 0, 1}    |
|     ------------------------------------------------------------------------|
|                                                                             |
|     7894p         p          - 3300p         p          - 850p         p    |
|          {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}       {1, 0, 0} {1, |
|     ------------------------------------------------------------------------|
|                                                                         2   |
|           - 10956p         p          + 6865p         p          + 8470p    |
|     1, 1}         {1, 0, 1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}        {1, |
|     ------------------------------------------------------------------------|
|                                         2                                   |
|          , p         p          + 12103p          - 15038p         p        |
|     1, 1}   {0, 1, 0} {1, 0, 0}         {1, 0, 0}         {0, 1, 1} {1, 0,  |
|     ------------------------------------------------------------------------|
|                                          2                                  |
|        + 3534p         p          + 7597p          + 7894p         p        |
|     1}        {1, 0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1, 0} {1, 1,  |
|     ------------------------------------------------------------------------|
|                                         2                                   |
|        - 850p         p          + 6865p          - 8785p         p         |
|     0}       {1, 0, 0} {1, 1, 0}        {1, 1, 0}        {0, 1, 0} {1, 1, 1}|
|     ------------------------------------------------------------------------|
|                                                                             |
|     + 14139p         p          + 5778p         p          + 9818p          |
|             {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {1, 0,    |
|     ------------------------------------------------------------------------|
|                                                      2                      |
|       p          + 11496p         p          - 13888p         , p           |
|     1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}         {1, 1, 1}   {0, 1,     |
|     ------------------------------------------------------------------------|
|                         2                                                   |
|       p          + 4731p          - 15131p         p          + 11961p      |
|     0} {0, 1, 1}        {0, 1, 1}         {0, 1, 1} {1, 0, 1}         {1, 0,|
|     ------------------------------------------------------------------------|
|                          2                                                  |
|       p          - 13703p          - 11806p         p          - 7596p      |
|     0} {1, 0, 1}         {1, 0, 1}         {0, 1, 0} {1, 1, 1}        {0, 1,|
|     ------------------------------------------------------------------------|
|                                                                             |
|       p          + 10561p         p          + 8010p         p          -   |
|     1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}     |
|     ------------------------------------------------------------------------|
|                                      2           2                 2        |
|     10049p         p          + 1546p         , p          - 12264p         |
|           {1, 1, 0} {1, 1, 1}        {1, 1, 1}   {0, 1, 0}         {0, 1, 1}|
|     ------------------------------------------------------------------------|
|             2                                                               |
|     + 11961p          - 8065p         p          - 10563p         p         |
|             {1, 0, 0}        {0, 1, 1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}|
|     ------------------------------------------------------------------------|
|            2                                                                |
|     - 8689p          - 11806p         p          + 10561p         p         |
|            {1, 0, 1}         {0, 1, 0} {1, 1, 0}         {1, 0, 0} {1, 1, 0}|
|     ------------------------------------------------------------------------|
|             2                                                               |
|     - 10049p          + 10273p         p          - 10313p         p        |
|             {1, 1, 0}         {0, 1, 0} {1, 1, 1}         {0, 1, 1} {1, 1,  |
|     ------------------------------------------------------------------------|
|                                                                             |
|        + 4458p         p          - 3454p         p          + 11487p       |
|     1}        {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}         {1, 1, |
|     ------------------------------------------------------------------------|
|                         2                                                   |
|       p          + 2116p         )                                          |
|     0} {1, 1, 1}        {1, 1, 1}                                           |
|                                                                             |
|o8 : Ideal of PR                                                             |
+-----------------------------------------------------------------------------+

Here is an example where the vertices of the graph need to be relabeled.

+-----------------------------------------------------------------------------+
|i9 : FF = ZZ/32003                                                           |
|                                                                             |
|o9 = FF                                                                      |
|                                                                             |
|o9 : QuotientRing                                                            |
+-----------------------------------------------------------------------------+
|i10 : d = {2,3,2};                                                           |
+-----------------------------------------------------------------------------+
|i11 : X = randomGame(d, CoefficientRing => FF);                              |
+-----------------------------------------------------------------------------+
|i12 : PR = probabilityRing(d, CoefficientRing => FF);                        |
+-----------------------------------------------------------------------------+
|i13 : G1 = graph {{John,Matthew},{Matthew,Sarah}};                           |
+-----------------------------------------------------------------------------+
|i14 : G2 = graph {{a,b},{b,c},{c,a}};                                        |
+-----------------------------------------------------------------------------+
|i15 : I1 = spohnCI(PR,X,G1, {John,Matthew,Sarah})                            |
|                                                                             |
|                                                                             |
|o15 = ideal (p         p          - p         p         , p         p        |
|              {0, 2, 1} {1, 2, 0}    {0, 2, 0} {1, 2, 1}   {0, 1, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|         - p         p         , p         p          - p         p         ,|
|      0}    {0, 1, 0} {1, 1, 1}   {0, 0, 1} {1, 0, 0}    {0, 0, 0} {1, 0, 1} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      p         p          - 15130p         p          - 8126p         p     |
|       {0, 0, 0} {1, 0, 0}         {0, 1, 0} {1, 0, 0}        {0, 1, 1} {1,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|            + 15935p         p          + 1045p         p          -         |
|      0, 0}         {0, 2, 0} {1, 0, 0}        {0, 2, 1} {1, 0, 0}           |
|      -----------------------------------------------------------------------|
|                                                                             |
|      15144p         p          - 15145p         p          - 14652p         |
|            {0, 0, 0} {1, 0, 1}         {0, 0, 1} {1, 0, 1}         {0, 1,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 7648p         p          - 15590p         p          +  |
|      0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}         {0, 2, 0} {1, 0, 1}    |
|      -----------------------------------------------------------------------|
|                                                                             |
|      1523p         p          - 3924p         p          + 12455p           |
|           {0, 2, 1} {1, 0, 1}        {0, 0, 0} {1, 1, 0}         {0, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 12948p         p          + 12010p         p          - |
|      1} {1, 1, 0}         {0, 1, 0} {1, 1, 0}         {0, 2, 0} {1, 1, 0}   |
|      -----------------------------------------------------------------------|
|                                                                             |
|      2880p         p          - 15478p         p          + 901p            |
|           {0, 2, 1} {1, 1, 0}         {0, 0, 0} {1, 1, 1}       {0, 0,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 10657p         p          + 8398p         p          +  |
|      1} {1, 1, 1}         {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                                                                             |
|      456p         p          - 14434p         p          + 10874p           |
|          {0, 2, 0} {1, 1, 1}         {0, 2, 1} {1, 1, 1}         {0, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 4750p         p          - 4257p         p          +   |
|      0} {1, 2, 0}        {0, 0, 1} {1, 2, 0}        {0, 1, 0} {1, 2, 0}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      2747p         p          - 5195p         p          - 1623p            |
|           {0, 1, 1} {1, 2, 0}        {0, 2, 0} {1, 2, 0}        {0, 0,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 14756p         p          + 15249p         p          - |
|      0} {1, 2, 1}         {0, 0, 1} {1, 2, 1}         {0, 1, 0} {1, 2, 1}   |
|      -----------------------------------------------------------------------|
|                                                                             |
|      9750p         p          - 5774p         p          - 579p         p   |
|           {0, 1, 1} {1, 2, 1}        {0, 2, 0} {1, 2, 1}       {0, 2, 1} {1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           , p         p          - 8647p         p          - 8198p         |
|      2, 1}   {0, 1, 0} {0, 2, 0}        {0, 1, 1} {0, 2, 0}        {0, 1,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 15157p         p          + 14818p         p          + |
|      0} {0, 2, 1}         {0, 1, 1} {0, 2, 1}         {0, 2, 0} {1, 1, 0}   |
|      -----------------------------------------------------------------------|
|                                                                             |
|      6619p         p          + 9296p         p          + 1097p            |
|           {0, 2, 1} {1, 1, 0}        {0, 2, 0} {1, 1, 1}        {0, 2,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 3959p         p          - 4689p         p          -   |
|      1} {1, 1, 1}        {0, 1, 0} {1, 2, 0}        {0, 1, 1} {1, 2, 0}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      13227p         p          + 13254p         p          + 12234p         |
|            {1, 1, 0} {1, 2, 0}         {1, 1, 1} {1, 2, 0}         {0, 1,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 3586p         p          - 4952p         p          -   |
|      0} {1, 2, 1}        {0, 1, 1} {1, 2, 1}        {1, 1, 0} {1, 2, 1}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      10474p         p         , p         p          - 1636p         p      |
|            {1, 1, 1} {1, 2, 1}   {0, 0, 0} {0, 2, 0}        {0, 0, 1} {0, 2,|
|      -----------------------------------------------------------------------|
|                                                                             |
|         - 5775p         p          - 7412p         p          - 10584p      |
|      0}        {0, 0, 0} {0, 2, 1}        {0, 0, 1} {0, 2, 1}         {0, 2,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 15643p         p          - 7986p         p          -  |
|      0} {1, 0, 0}         {0, 2, 1} {1, 0, 0}        {0, 2, 0} {1, 0, 1}    |
|      -----------------------------------------------------------------------|
|                                                                             |
|      13762p         p          - 15076p         p          + 15290p         |
|            {0, 2, 1} {1, 0, 1}         {0, 0, 0} {1, 2, 0}         {0, 0,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 6342p         p          + 8940p         p          +   |
|      1} {1, 2, 0}        {1, 0, 0} {1, 2, 0}        {1, 0, 1} {1, 2, 0}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      9622p         p          + 7985p         p          - 963p         p   |
|           {0, 0, 0} {1, 2, 1}        {0, 0, 1} {1, 2, 1}       {1, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|            + 1635p         p         , p         p          - 13265p        |
|      2, 1}        {1, 0, 1} {1, 2, 1}   {0, 0, 0} {0, 1, 0}         {0, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 342p         p          - 12924p         p          -   |
|      1} {0, 1, 0}       {0, 0, 0} {0, 1, 1}         {0, 0, 1} {0, 1, 1}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      2242p         p          - 1901p         p          + 4951p            |
|           {0, 1, 0} {1, 0, 0}        {0, 1, 1} {1, 0, 0}        {0, 1,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 5292p         p          - 6804p         p          +   |
|      0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}        {0, 0, 0} {1, 1, 0}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      11933p         p          - 9047p         p          - 1854p           |
|            {0, 0, 1} {1, 1, 0}        {1, 0, 0} {1, 1, 0}        {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 8592p         p          + 10145p         p          -  |
|      1} {1, 1, 0}        {0, 0, 0} {1, 1, 1}         {0, 0, 1} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                                                                             |
|      10835p         p          - 3642p         p         , p         p      |
|            {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}   {0, 0, 0} {0, 0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|         - 13133p         p          + 2564p         p          - 10570p     |
|      1}         {0, 0, 1} {0, 1, 0}        {0, 0, 0} {0, 1, 1}         {0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p          - 12449p         p          - 9886p         p          |
|      1, 0} {0, 1, 1}         {0, 0, 1} {0, 2, 0}        {0, 1, 1} {0, 2, 0} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      - 7579p         p          + 11290p         p          + 11974p        |
|             {0, 0, 0} {0, 2, 1}         {0, 1, 0} {0, 2, 1}         {0, 2,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 3900p         p          - 6243p         p          -   |
|      0} {0, 2, 1}        {0, 1, 1} {1, 0, 0}        {0, 2, 1} {1, 0, 0}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      402p         p          - 14873p         p          - 14189p           |
|          {0, 0, 0} {1, 0, 1}         {0, 1, 0} {1, 0, 1}         {0, 2,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 403p         p          + 4755p         p          -    |
|      0} {1, 0, 1}       {1, 0, 0} {1, 0, 1}        {0, 0, 1} {1, 1, 0}      |
|      -----------------------------------------------------------------------|
|                                                                             |
|      2825p         p          + 3015p         p          - 3204p            |
|           {0, 2, 1} {1, 1, 0}        {1, 0, 1} {1, 1, 0}        {0, 0,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 9020p         p          - 15654p         p          -  |
|      0} {1, 1, 1}        {0, 1, 0} {1, 1, 1}         {0, 2, 0} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                                                                             |
|      1868p         p          + 1550p         p          + 5231p            |
|           {1, 0, 0} {1, 1, 1}        {1, 1, 0} {1, 1, 1}        {0, 0,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 7794p         p          + 3491p         p          +   |
|      1} {1, 2, 0}        {0, 1, 1} {1, 2, 0}        {1, 0, 1} {1, 2, 0}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      2026p         p          + 4832p         p          - 8302p            |
|           {1, 1, 1} {1, 2, 0}        {0, 0, 0} {1, 2, 1}        {0, 1,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 9967p         p          + 6168p         p          +   |
|      0} {1, 2, 1}        {0, 2, 0} {1, 2, 1}        {1, 0, 0} {1, 2, 1}     |
|      -----------------------------------------------------------------------|
|                                                                             |
|      9586p         p          + 10062p         p         , p         p      |
|           {1, 1, 0} {1, 2, 1}         {1, 2, 0} {1, 2, 1}   {0, 1, 0} {1, 0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 1971p         p         p           |
|      0} {1, 0, 1} {1, 1, 0} {1, 2, 0}        {0, 2, 0} {1, 0, 0} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          + 3758p         p         p         p          -|
|      1} {1, 1, 0} {1, 2, 0}        {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|      12981p         p         p         p          - 10119p         p       |
|            {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}         {0, 0, 1} {1, 0, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 3239p         p         p         p          -|
|      1} {1, 1, 0} {1, 2, 0}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|                     2                                              2        |
|      6694p         p         p         p          + 5036p         p         |
|           {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}        {1, 0, 0} {1, 0,   |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p          + 3035p         p         p         p          -|
|      1} {1, 1, 0} {1, 2, 0}        {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|                                2                                            |
|      12186p         p         p         p          + 13376p         p       |
|            {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 2, 0}         {0, 1, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p         p          - 13318p         p         p         p          |
|      1} {1, 1, 0} {1, 2, 0}         {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|                                  2                          2         2     |
|      - 10548p         p         p         p          + 5631p         p      |
|              {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}        {1, 0, 1} {1, 1,|
|      -----------------------------------------------------------------------|
|                         2                                              2    |
|        p          + 952p         p         p         p          - 6519p     |
|      0} {1, 2, 0}       {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 0}        {0,  |
|      -----------------------------------------------------------------------|
|                                                            2                |
|           p         p         p          - 13515p         p         p       |
|      2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 0}         {0, 1, 0} {1, 0, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          - 15944p         p         p         p          - 1705p   |
|      1} {1, 2, 0}         {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 0}        {0,|
|      -----------------------------------------------------------------------|
|            2                                    2                           |
|           p         p         p          - 6421p         p         p        |
|      2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 6452p         p         p         p         p          -|
|      1} {1, 2, 0}        {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|            2                                                                |
|      11220p         p         p         p          - 11694p         p       |
|            {0, 1, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}         {0, 0, 0} {0, 1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 957p         p         p            |
|      1} {1, 0, 1} {1, 1, 1} {1, 2, 0}       {0, 0, 1} {0, 1, 1} {1, 0,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 2883p         p         p         p           |
|      1} {1, 1, 1} {1, 2, 0}        {0, 1, 0} {0, 1, 1} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                         2                                                   |
|        p          + 896p         p         p         p          + 5561p     |
|      1} {1, 2, 0}       {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p         p          - 63p         p          |
|      0, 1} {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}      {0, 1, 1} {0, 2,    |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p         p          + 8577p         p         p           |
|      0} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, 2, 0} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 6518p         p         p         p         p          +|
|      1} {1, 2, 0}        {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      6895p         p         p         p         p          - 9802p         |
|           {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, 2,   |
|      -----------------------------------------------------------------------|
|                                                         2                   |
|        p         p         p         p          - 15218p         p          |
|      0} {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}         {1, 0, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          - 12981p         p         p         p          |
|      1} {1, 1, 1} {1, 2, 0}         {0, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|                        2                                              2     |
|      + 10883p         p         p         p          - 6023p         p      |
|              {0, 0, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, 1, 0} {1, 0,|
|      -----------------------------------------------------------------------|
|                                            2                                |
|        p         p          + 86p         p         p         p          +  |
|      1} {1, 1, 1} {1, 2, 0}      {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0}    |
|      -----------------------------------------------------------------------|
|                     2                                              2        |
|      7976p         p         p         p          - 9360p         p         |
|           {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 0,   |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p         p          - 13748p         p         p         p          |
|      1} {1, 1, 1} {1, 2, 0}         {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|                                                                   2         |
|      - 99p         p         p         p         p          + 854p          |
|           {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}       {0, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 3916p         p         p           |
|      0} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {0, 2, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p         p          + 13202p         p         p         p          |
|      0} {1, 1, 1} {1, 2, 0}         {0, 2, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      - 12179p         p         p         p         p          + 10311p     |
|              {0, 1, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {0,  |
|      -----------------------------------------------------------------------|
|                                                            2                |
|           p         p         p         p          - 14794p         p       |
|      2, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {1, 0, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 4537p         p         p         p           |
|      0} {1, 1, 1} {1, 2, 0}        {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 11914p         p         p         p         p          |
|      1} {1, 2, 0}         {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      - 9499p         p         p         p         p          - 2081p       |
|             {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          - 11758p         p          |
|      0} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {1, 0, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p         p          - 12321p         p         p          |
|      1} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {1, 0, 1} {1, 1, 0} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p          + 8621p         p         p         p          + 13930p   |
|      1} {1, 2, 0}        {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {0,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p         p          + 15980p         p         p       |
|      1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {0, 2, 0} {1, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          - 12944p         p         p         p          +         |
|      1} {1, 2, 0}         {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}           |
|      -----------------------------------------------------------------------|
|                      2                                    2         2       |
|      12748p         p         p         p          - 2084p         p        |
|            {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                               2                          2  |
|        p          - 12041p         p         p         p          + 1860p   |
|      1} {1, 2, 0}         {0, 0, 1} {0, 1, 0} {1, 1, 1} {1, 2, 0}        {0,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          - 13202p         p         p         p       |
|      1, 0} {1, 1, 1} {1, 2, 0}         {0, 0, 1} {0, 1, 1} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                                     2                          2         2  |
|         + 11741p         p         p         p          - 2189p         p   |
|      0}         {0, 1, 0} {0, 1, 1} {1, 1, 1} {1, 2, 0}        {0, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|                                                2                            |
|           p          + 613p         p         p         p          +        |
|      1, 1} {1, 2, 0}       {0, 0, 1} {0, 2, 0} {1, 1, 1} {1, 2, 0}          |
|      -----------------------------------------------------------------------|
|                                2                          2         2       |
|      10721p         p         p         p          + 3555p         p        |
|            {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2, 0}        {0, 2, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          + 9374p         p         p         p          + 12449p   |
|      1} {1, 2, 0}        {0, 1, 0} {1, 0, 0} {1, 1, 1} {1, 2, 0}         {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p         p          + 6110p         p         p        |
|      1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 0}        {0, 2, 0} {1, 0, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                          2         2                                        |
|        p          - 3988p         p         p          - 14937p         p   |
|      1} {1, 2, 0}        {1, 0, 0} {1, 1, 1} {1, 2, 0}         {0, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          - 5793p         p         p         p        |
|      0, 1} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {1, 0, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         + 4783p         p         p         p          + 1469p         p    |
|      0}        {0, 1, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          - 12032p         p         p         p       |
|      0, 1} {1, 1, 1} {1, 2, 0}         {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                                    2                          2         2   |
|         + 4230p         p         p         p          + 8337p         p    |
|      0}        {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {1, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p          - 3491p         p         p         p          -       |
|      1, 1} {1, 2, 0}        {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}         |
|      -----------------------------------------------------------------------|
|                               2                                             |
|      1605p         p         p         p          + 11119p         p        |
|           {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {0, 2, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p         p          + 3331p         p         p         p          +|
|      0} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|                               2                          2         2        |
|      6762p         p         p         p          + 5678p         p         |
|           {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {1, 1, 0} {1, 1,   |
|      -----------------------------------------------------------------------|
|                                    3                                     3  |
|        p          - 2943p         p         p          - 14456p         p   |
|      1} {1, 2, 0}        {0, 0, 1} {1, 1, 1} {1, 2, 0}         {0, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|                                        3                                    |
|           p          - 14340p         p         p          + 10490p         |
|      1, 1} {1, 2, 0}         {0, 1, 1} {1, 1, 1} {1, 2, 0}         {0, 2,   |
|      -----------------------------------------------------------------------|
|         3                                    3                              |
|        p         p          + 7808p         p         p          - 12716p   |
|      0} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 1, 1} {1, 2, 0}         {1,|
|      -----------------------------------------------------------------------|
|            3                                    3                           |
|           p         p          + 8173p         p         p          -       |
|      0, 1} {1, 1, 1} {1, 2, 0}        {1, 1, 0} {1, 1, 1} {1, 2, 0}         |
|      -----------------------------------------------------------------------|
|                                         2                                   |
|      4773p         p         p         p          + 12009p         p        |
|           {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}         {0, 0, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p          + 5568p         p         p         p          +|
|      1} {1, 1, 0} {1, 2, 0}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      10877p         p         p         p          - 12104p         p       |
|            {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}         {1, 0, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|                   2                 2                   2                2  |
|        p         p          - 14111p         p         p          - 1232p   |
|      1} {1, 1, 0} {1, 2, 0}         {1, 0, 1} {1, 1, 0} {1, 2, 0}        {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          + 7165p         p         p         p        |
|      0, 1} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {0, 1, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                2                   2                                        |
|         - 1482p         p         p          - 5598p         p         p    |
|      0}        {0, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {0, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p          - 15847p         p         p         p          +      |
|      1, 1} {1, 2, 0}         {0, 1, 0} {0, 1, 1} {1, 1, 1} {1, 2, 0}        |
|      -----------------------------------------------------------------------|
|           2                   2                                             |
|      4097p         p         p          - 11p         p         p           |
|           {0, 1, 1} {1, 1, 1} {1, 2, 0}      {0, 0, 1} {0, 2, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|         2                                               2                2  |
|        p          + 12906p         p         p         p          + 5188p   |
|      1} {1, 2, 0}         {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2, 0}        {0,|
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p          + 12328p         p         p         p       |
|      2, 0} {1, 1, 1} {1, 2, 0}         {0, 1, 0} {1, 0, 0} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|         + 2806p         p         p         p          - 11202p         p   |
|      0}        {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 0}         {0, 2, 0} {1,|
|      -----------------------------------------------------------------------|
|                      2                 2                   2                |
|           p         p          - 10281p         p         p          +      |
|      0, 0} {1, 1, 1} {1, 2, 0}         {1, 0, 0} {1, 1, 1} {1, 2, 0}        |
|      -----------------------------------------------------------------------|
|                                         2                                   |
|      5860p         p         p         p          - 12864p         p        |
|           {0, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}         {0, 0, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p          + 5734p         p         p         p          +|
|      1} {1, 1, 1} {1, 2, 0}        {0, 1, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      14320p         p         p         p          - 12334p         p       |
|            {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0}         {0, 2, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|                   2                                               2         |
|        p         p          - 14370p         p         p         p          |
|      1} {1, 1, 1} {1, 2, 0}         {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|             2                   2                                           |
|      + 2363p         p         p          + 493p         p         p        |
|             {1, 0, 1} {1, 1, 1} {1, 2, 0}       {0, 0, 1} {1, 1, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p          - 14718p         p         p         p          + 5862p   |
|      1} {1, 2, 0}         {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {0,|
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          - 7630p         p         p        |
|      2, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 1, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|         2                                             2                 2   |
|        p          + 324p         p         p         p          + 13074p    |
|      1} {1, 2, 0}       {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {1, |
|      -----------------------------------------------------------------------|
|                      2                          2         2                 |
|           p         p          + 8956p         p         p          -       |
|      1, 0} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {1, 1, 1} {1, 2, 0}         |
|      -----------------------------------------------------------------------|
|                      2         2                           2         2      |
|      15028p         p         p          - 13714p         p         p       |
|            {0, 1, 0} {1, 1, 1} {1, 2, 0}         {0, 1, 1} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                          2         2                          2         2   |
|         - 1310p         p         p          - 4951p         p         p    |
|      0}        {0, 2, 0} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|                             2         2                           2         |
|            + 6691p         p         p          - 11854p         p          |
|      2, 0}        {1, 0, 1} {1, 1, 1} {1, 2, 0}         {1, 1, 0} {1, 1,    |
|      -----------------------------------------------------------------------|
|         2                 3         2                                   3   |
|        p          - 11302p         p          - 402p         p         p    |
|      1} {1, 2, 0}         {1, 1, 1} {1, 2, 0}       {0, 0, 1} {1, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|                                       3                                     |
|            + 2248p         p         p          + 9461p         p           |
|      2, 0}        {0, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 1, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|         3                                    3                              |
|        p          + 1044p         p         p          - 9115p         p    |
|      1} {1, 2, 0}        {0, 2, 0} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, |
|      -----------------------------------------------------------------------|
|            3                                    3                           |
|           p          - 1609p         p         p          - 9545p           |
|      1, 1} {1, 2, 0}        {1, 0, 1} {1, 1, 1} {1, 2, 0}        {1, 1,     |
|      -----------------------------------------------------------------------|
|                   3                2         3                              |
|        p         p          - 9577p         p          + 5571p         p    |
|      0} {1, 1, 1} {1, 2, 0}        {1, 1, 1} {1, 2, 0}        {0, 2, 0} {0, |
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p         p         p          - 6841p         p         p        |
|      2, 1} {1, 0, 0} {1, 1, 0} {1, 2, 1}        {0, 2, 1} {1, 0, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p          + 6131p         p         p         p          + 15446p   |
|      0} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 1, 0} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2                                     2                          |
|           p         p         p          + 13719p         p         p       |
|      2, 1} {1, 0, 0} {1, 1, 0} {1, 2, 1}         {0, 0, 1} {1, 0, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 7678p         p         p         p         p          +|
|      0} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                      2      |
|      13994p         p         p         p         p          + 15286p       |
|            {0, 0, 1} {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}         {0, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 2151p         p         p           |
|      0} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 0, 0} {0, 2, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 2232p         p         p         p           |
|      1} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {0, 2, 1} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 14153p         p         p         p         p          |
|      0} {1, 2, 1}         {0, 1, 0} {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                                                      2      |
|      + 2539p         p         p         p         p          + 6751p       |
|             {0, 2, 0} {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 15613p         p         p          |
|      1} {1, 0, 1} {1, 1, 0} {1, 2, 1}         {0, 1, 0} {1, 0, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 15095p         p         p         p          |
|      1} {1, 1, 0} {1, 2, 1}         {0, 2, 0} {1, 0, 0} {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 2933p         p         p         p         p          -|
|      0} {1, 2, 1}        {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|           2                                                         2       |
|      4079p         p         p         p          - 12981p         p        |
|           {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}         {0, 0, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          + 11362p         p         p         p          |
|      1} {1, 1, 0} {1, 2, 1}         {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|      - 508p         p         p         p          - 3481p         p        |
|            {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 2, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          + 15700p         p         p         p          |
|      1} {1, 1, 0} {1, 2, 1}         {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                        2                                    2         2     |
|      - 10563p         p         p         p          - 3337p         p      |
|              {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {1, 1,|
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          + 7654p         p         p         p          - 7155p    |
|      0} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                      2                          2         2                 |
|           p         p         p          + 4803p         p         p        |
|      0, 1} {0, 2, 0} {1, 1, 0} {1, 2, 1}        {0, 2, 0} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         - 5651p         p         p         p          - 11112p         p   |
|      1}        {0, 0, 1} {0, 2, 1} {1, 1, 0} {1, 2, 1}         {0, 1, 0} {0,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          + 13787p         p         p         p       |
|      2, 1} {1, 1, 0} {1, 2, 1}         {0, 2, 0} {0, 2, 1} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                2         2                                               2  |
|         - 2890p         p         p          + 13122p         p         p   |
|      1}        {0, 2, 1} {1, 1, 0} {1, 2, 1}         {0, 2, 0} {1, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|                                                  2                          |
|           p          - 14745p         p         p         p          -      |
|      1, 0} {1, 2, 1}         {0, 2, 1} {1, 0, 0} {1, 1, 0} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                               2                                             |
|      6121p         p         p         p          - 12168p         p        |
|           {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}         {0, 0, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p         p          - 5403p         p         p         p          +|
|      1} {1, 1, 0} {1, 2, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                               2                                             |
|      9570p         p         p         p          + 3529p         p         |
|           {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 2, 1} {1, 0,   |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p         p          - 3892p         p         p         p          -|
|      1} {1, 1, 0} {1, 2, 1}        {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|            2         2                                   3                  |
|      12333p         p         p          + 612p         p         p         |
|            {1, 0, 1} {1, 1, 0} {1, 2, 1}       {0, 0, 1} {1, 1, 0} {1, 2, 1}|
|      -----------------------------------------------------------------------|
|                        3                                    3               |
|      - 14105p         p         p          + 3087p         p         p      |
|              {0, 2, 0} {1, 1, 0} {1, 2, 1}        {0, 2, 1} {1, 1, 0} {1, 2,|
|      -----------------------------------------------------------------------|
|                           3                           2                     |
|         + 14219p         p         p          - 12419p         p            |
|      1}         {1, 0, 1} {1, 1, 0} {1, 2, 1}         {0, 1, 1} {1, 0,      |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          - 7214p         p         p         p          -|
|      0} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      6433p         p         p         p         p          - 6217p         |
|           {0, 1, 0} {0, 2, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1}        {0, 1,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          + 11475p         p          |
|      1} {0, 2, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1}         {0, 2, 0} {0, 2,    |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p         p          - 15650p         p         p          |
|      1} {1, 0, 0} {1, 1, 1} {1, 2, 1}         {0, 2, 1} {1, 0, 0} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          + 14576p         p         p         p          - 3389p   |
|      1} {1, 2, 1}         {0, 1, 0} {1, 0, 0} {1, 1, 1} {1, 2, 1}        {0,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p         p          - 3414p         p         p        |
|      1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          - 10216p         p         p         p          +         |
|      1} {1, 2, 1}         {0, 2, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1}           |
|      -----------------------------------------------------------------------|
|            2                                                                |
|      10811p         p         p         p          + 6873p         p        |
|            {0, 0, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 1,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p         p          + 6368p         p         p           |
|      0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 0} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 15729p         p         p         p         p          |
|      1} {1, 2, 1}         {0, 0, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      + 10594p         p         p         p         p          - 9464p      |
|              {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 1,|
|      -----------------------------------------------------------------------|
|                                                         2                   |
|        p         p         p         p          - 13571p         p          |
|      0} {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0, 1, 1} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 3389p         p         p         p           |
|      1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 2, 0} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 13593p         p         p         p         p          |
|      1} {1, 2, 1}         {0, 1, 1} {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|             2                                                               |
|      - 7338p         p         p         p          - 3304p         p       |
|             {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 0, 0} {0, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 7620p         p         p           |
|      1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 2, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 13323p         p         p         p          |
|      1} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {0, 2, 1} {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 13135p         p         p         p         p          |
|      1} {1, 2, 1}         {0, 1, 1} {0, 2, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                                                      2      |
|      - 8065p         p         p         p         p          - 7343p       |
|             {0, 2, 0} {0, 2, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 14639p         p         p          |
|      1} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, 0, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 3004p         p         p         p           |
|      1} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 7091p         p         p         p         p          +|
|      1} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                    2        |
|      7658p         p         p         p         p          - 8287p         |
|           {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {1, 0,   |
|      -----------------------------------------------------------------------|
|                                                         2                   |
|        p         p         p          - 12981p         p         p          |
|      0} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0, 0, 0} {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p          - 3733p         p         p         p          + 13053p   |
|      1} {1, 2, 1}        {0, 0, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p         p          - 2739p         p         p        |
|      1, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          + 13679p         p         p         p          -         |
|      1} {1, 2, 1}         {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}           |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|      14246p         p         p         p          - 2584p         p        |
|            {0, 2, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {1, 0, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          + 9423p         p         p         p          -|
|      1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                     2       |
|      4723p         p         p         p         p          + 10655p        |
|           {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 1,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 5122p         p         p           |
|      0} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 2, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p         p          + 13701p         p         p         p          |
|      0} {1, 1, 1} {1, 2, 1}         {0, 2, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      - 5557p         p         p         p         p          + 4297p       |
|             {0, 0, 1} {0, 2, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          + 7517p         p           |
|      0} {0, 2, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {0, 2,     |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p         p          + 10590p         p         p          |
|      1} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 2, 1} {1, 1, 0} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 4461p         p         p         p         p          -|
|      1} {1, 2, 1}        {0, 1, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      6850p         p         p         p         p          - 10884p        |
|           {0, 2, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 2,  |
|      -----------------------------------------------------------------------|
|                                                         2                   |
|        p         p         p         p          - 11165p         p          |
|      1} {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {1, 0, 0} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 13583p         p         p         p          |
|      0} {1, 1, 1} {1, 2, 1}         {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 14405p         p         p         p         p          |
|      1} {1, 2, 1}         {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      + 3895p         p         p         p         p          + 1256p       |
|             {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          + 1274p         p           |
|      0} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 4356p         p         p           |
|      1} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {1, 0, 0} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                   2                                         |
|        p         p          - 491p         p         p         p          + |
|      0} {1, 1, 1} {1, 2, 1}       {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}   |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|      11076p         p         p         p          + 13918p         p       |
|            {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 6487p         p         p         p          +|
|      0} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|      15248p         p         p         p          + 10952p         p       |
|            {0, 2, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {1, 0, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 9138p         p         p         p          -|
|      0} {1, 1, 1} {1, 2, 1}        {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|            3                                    2         2                 |
|      10098p         p         p          - 5647p         p         p        |
|            {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                     2                           2           |
|         + 15933p         p         p         p          + 12177p            |
|      1}         {0, 0, 1} {0, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 1,      |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p         p          - 6347p         p         p         p          +|
|      0} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 1, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                               2                          2         2        |
|      9316p         p         p         p          - 8456p         p         |
|           {0, 1, 0} {0, 1, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {1, 1,   |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p          + 11717p         p         p         p          -         |
|      1} {1, 2, 1}         {0, 0, 1} {0, 2, 0} {1, 1, 1} {1, 2, 1}           |
|      -----------------------------------------------------------------------|
|                                2                           2         2      |
|      13797p         p         p         p          + 10006p         p       |
|            {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2, 1}         {0, 2, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          - 7464p         p         p         p          + 6071p    |
|      1} {1, 2, 1}        {0, 0, 1} {0, 2, 1} {1, 1, 1} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p         p          - 9580p         p         p        |
|      1, 0} {0, 2, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {0, 2, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                              2                          2   |
|        p          + 3934p         p         p         p          + 8528p    |
|      1} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 1, 1} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|            2                                             2                  |
|           p         p          - 951p         p         p         p         |
|      2, 1} {1, 1, 1} {1, 2, 1}       {0, 1, 0} {1, 0, 0} {1, 1, 1} {1, 2, 1}|
|      -----------------------------------------------------------------------|
|                                 2                                           |
|      + 8696p         p         p         p          + 10103p         p      |
|             {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1}         {0, 2, 0} {1, 0,|
|      -----------------------------------------------------------------------|
|         2                                             2                     |
|        p         p          + 988p         p         p         p          - |
|      0} {1, 1, 1} {1, 2, 1}       {0, 2, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1}   |
|      -----------------------------------------------------------------------|
|           2         2                                              2        |
|      7421p         p         p          + 3605p         p         p         |
|           {1, 0, 0} {1, 1, 1} {1, 2, 1}        {0, 0, 0} {1, 0, 1} {1, 1,   |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          - 9300p         p         p         p          + 5346p    |
|      1} {1, 2, 1}        {0, 0, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p         p          + 6934p         p         p        |
|      1, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          - 8145p         p         p         p          - 13531p   |
|      1} {1, 2, 1}        {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p         p          + 4494p         p         p        |
|      2, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {1, 0, 0} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                          2         2                                        |
|        p          - 9495p         p         p          + 14525p         p   |
|      1} {1, 2, 1}        {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          - 4785p         p         p         p        |
|      1, 0} {1, 1, 1} {1, 2, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         - 1971p         p         p         p          + 13295p         p   |
|      1}        {0, 2, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 2, 1} {1,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          - 5277p         p         p         p        |
|      1, 0} {1, 1, 1} {1, 2, 1}        {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                     2                          2         2  |
|         + 10235p         p         p         p          + 2060p         p   |
|      1}         {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {1, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|                                       3                                     |
|           p          + 1480p         p         p          + 7756p           |
|      1, 1} {1, 2, 1}        {0, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 1,     |
|      -----------------------------------------------------------------------|
|         3                                    3                              |
|        p         p          - 5968p         p         p          - 5792p    |
|      0} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {1, 1, 1} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|            3                                    3                           |
|           p         p          + 3500p         p         p          -       |
|      2, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 1, 1} {1, 2, 1}         |
|      -----------------------------------------------------------------------|
|                      3                                     3                |
|      13201p         p         p          - 13212p         p         p       |
|            {1, 0, 0} {1, 1, 1} {1, 2, 1}         {1, 0, 1} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                          3                          2                       |
|         - 3668p         p         p          - 3558p         p         p    |
|      1}        {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {1, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p          - 14690p         p         p         p         p       |
|      2, 0} {1, 2, 1}         {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 2, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|         + 12927p         p         p         p         p          +         |
|      1}         {0, 0, 1} {0, 2, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}           |
|      -----------------------------------------------------------------------|
|            2                                                                |
|      11579p         p         p         p          + 13843p         p       |
|            {0, 2, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}         {0, 1, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 1301p         p         p           |
|      0} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          + 9075p         p         p         p          -|
|      0} {1, 2, 0} {1, 2, 1}        {1, 0, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      9913p         p         p         p         p          - 11391p        |
|           {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}         {0, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          + 10760p         p          |
|      1} {1, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}         {0, 1, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 3560p         p         p           |
|      1} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, 2, 0} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 7315p         p         p         p           |
|      0} {1, 2, 0} {1, 2, 1}        {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2,     |
|      -----------------------------------------------------------------------|
|                          2                                                  |
|        p          + 1640p         p         p         p          + 3817p    |
|      0} {1, 2, 1}        {1, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p         p          + 8193p         p         p        |
|      0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          - 14701p         p         p         p          + 6259p   |
|      0} {1, 2, 1}         {0, 2, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {1,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p         p          + 5845p         p         p        |
|      0, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {1, 0, 1} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                          3                                     2            |
|        p          - 8901p         p         p          - 11890p         p   |
|      0} {1, 2, 1}        {1, 1, 0} {1, 2, 0} {1, 2, 1}         {0, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p          - 8561p         p         p         p        |
|      1, 1} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                           2                                                 |
|        p          - 14415p         p         p         p          + 5340p   |
|      0} {1, 2, 1}         {0, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p         p          - 10752p         p       |
|      0, 1} {0, 1, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 1, 0} {0, 1, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p         p          + 9653p         p         p           |
|      1} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 1, 1} {1, 1, 1} {1, 2,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 1614p         p         p         p         p          -|
|      0} {1, 2, 1}        {0, 0, 1} {0, 2, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                     2       |
|      7410p         p         p         p         p          + 13457p        |
|           {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 2,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 8139p         p         p           |
|      0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {1, 0, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 4502p         p         p         p           |
|      1} {1, 2, 0} {1, 2, 1}        {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 5603p         p         p         p         p          -|
|      0} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|           2                                                                 |
|      6694p         p         p         p          - 15449p         p        |
|           {1, 0, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 0, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 5909p         p         p           |
|      1} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 558p         p         p         p            |
|      1} {1, 2, 0} {1, 2, 1}       {0, 1, 0} {1, 0, 1} {1, 1, 1} {1, 2,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 9526p         p         p         p         p          +|
|      0} {1, 2, 1}        {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      9121p         p         p         p         p          + 6158p         |
|           {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {1, 0,   |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p         p          + 2026p         p           |
|      0} {1, 0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 14059p         p         p         p          |
|      1} {1, 2, 0} {1, 2, 1}         {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 5548p         p         p         p         p          +|
|      0} {1, 2, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      9035p         p         p         p         p          - 7832p         |
|           {0, 2, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {1, 0,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          + 907p         p            |
|      0} {1, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}       {1, 0, 1} {1, 1,      |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p         p          + 15851p         p         p          |
|      0} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {1, 1, 0} {1, 1, 1} {1, 2,    |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          - 13600p         p         p         p          + 799p    |
|      0} {1, 2, 1}         {0, 0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}       {0, |
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p         p          + 5689p         p         p        |
|      1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 1, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          + 11472p         p         p         p          -         |
|      0} {1, 2, 1}         {0, 2, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}           |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|      13322p         p         p         p          + 6856p         p        |
|            {1, 0, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 4007p         p         p         p          -|
|      1} {1, 2, 0} {1, 2, 1}        {1, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|           3                                                       2         |
|      1373p         p         p          - 337p         p         p          |
|           {1, 1, 1} {1, 2, 0} {1, 2, 1}       {0, 0, 1} {1, 1, 0} {1, 2,    |
|      -----------------------------------------------------------------------|
|                                             2                               |
|        p          - 877p         p         p         p          + 11361p    |
|      0} {1, 2, 1}       {0, 1, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}         {0, |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p         p          + 4739p         p         p        |
|      2, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {1, 0, 0} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                              2                          2   |
|        p          - 8272p         p         p         p          + 8851p    |
|      0} {1, 2, 1}        {1, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {1, |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          - 11024p         p         p         p       |
|      1, 0} {1, 2, 0} {1, 2, 1}         {0, 0, 1} {1, 1, 1} {1, 2, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         + 1315p         p         p         p          + 10974p         p   |
|      1}        {0, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          + 12410p         p         p         p       |
|      1, 1} {1, 2, 0} {1, 2, 1}         {0, 2, 0} {1, 1, 1} {1, 2, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|         - 14859p         p         p         p          - 4p         p      |
|      1}         {1, 0, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}     {1, 0, 1} {1, 1,|
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p         p          + 15071p         p         p         p          |
|      1} {1, 2, 0} {1, 2, 1}         {1, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|              2         2                                     3              |
|      + 15269p         p         p          - 12999p         p         p     |
|              {1, 1, 1} {1, 2, 0} {1, 2, 1}         {1, 1, 1} {1, 2, 0} {1,  |
|      -----------------------------------------------------------------------|
|                   2                   2                                     |
|            + 3534p         p         p          + 2908p         p           |
|      2, 1}        {0, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {0, 1,     |
|      -----------------------------------------------------------------------|
|                   2                                               2         |
|        p         p          + 14939p         p         p         p          |
|      0} {1, 1, 0} {1, 2, 1}         {0, 0, 1} {0, 2, 0} {1, 1, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|             2                   2                                           |
|      + 7248p         p         p          - 7250p         p         p       |
|             {0, 2, 0} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {0, 2, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p          - 1905p         p         p         p          + 15109p   |
|      0} {1, 2, 1}        {0, 1, 0} {0, 2, 1} {1, 1, 0} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|                                2                2                   2       |
|           p         p         p          + 3086p         p         p        |
|      2, 0} {0, 2, 1} {1, 1, 0} {1, 2, 1}        {0, 2, 1} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                               2                             |
|         - 14331p         p         p         p          - 11967p            |
|      1}         {0, 1, 0} {1, 0, 0} {1, 1, 0} {1, 2, 1}         {0, 2,      |
|      -----------------------------------------------------------------------|
|                             2                                               |
|        p         p         p          + 7349p         p         p           |
|      0} {1, 0, 0} {1, 1, 0} {1, 2, 1}        {0, 2, 1} {1, 0, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|         2                2                   2                              |
|        p          + 9603p         p         p          + 222p         p     |
|      0} {1, 2, 1}        {1, 0, 0} {1, 1, 0} {1, 2, 1}       {0, 0, 0} {1,  |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          + 4603p         p         p         p        |
|      0, 1} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|         - 3167p         p         p         p          - 6608p         p    |
|      1}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 2, 0} {1, |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p          + 13829p         p         p         p       |
|      0, 1} {1, 1, 0} {1, 2, 1}         {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                              2                 2            |
|         - 6312p         p         p         p          + 13714p         p   |
|      1}        {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}         {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|            2                           2         2                          |
|           p          - 10579p         p         p          - 1263p          |
|      1, 0} {1, 2, 1}         {0, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 1,    |
|      -----------------------------------------------------------------------|
|         2         2                          2         2                    |
|        p         p          - 4364p         p         p          + 14766p   |
|      0} {1, 1, 0} {1, 2, 1}        {0, 2, 0} {1, 1, 0} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2         2                           2         2                |
|           p         p          + 14853p         p         p          +      |
|      2, 1} {1, 1, 0} {1, 2, 1}         {1, 0, 0} {1, 1, 0} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                     2         2               3         2                   |
|      1572p         p         p          - 581p         p          +         |
|           {1, 0, 1} {1, 1, 0} {1, 2, 1}       {1, 1, 0} {1, 2, 1}           |
|      -----------------------------------------------------------------------|
|            2                   2                                            |
|      14395p         p         p          - 667p         p         p         |
|            {0, 0, 1} {1, 1, 1} {1, 2, 1}       {0, 0, 1} {0, 1, 0} {1, 1,   |
|      -----------------------------------------------------------------------|
|         2                2                   2                              |
|        p          + 8089p         p         p          - 11617p         p   |
|      1} {1, 2, 1}        {0, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          - 7144p         p         p         p        |
|      1, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 0} {0, 1, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                2                   2                                        |
|         - 1676p         p         p          - 13512p         p         p   |
|      1}        {0, 1, 1} {1, 1, 1} {1, 2, 1}         {0, 0, 1} {0, 2, 0} {1,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p          + 15630p         p         p         p          -      |
|      1, 1} {1, 2, 1}         {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|            2                   2                                            |
|      10644p         p         p          - 1584p         p         p        |
|            {0, 2, 0} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 2, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p          - 14201p         p         p         p          - 9473p   |
|      1} {1, 2, 1}         {0, 1, 0} {0, 2, 1} {1, 1, 1} {1, 2, 1}        {0,|
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          + 7230p         p         p        |
|      1, 1} {0, 2, 1} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|         2                 2                   2                             |
|        p          + 14643p         p         p          + 13885p            |
|      1} {1, 2, 1}         {0, 2, 1} {1, 1, 1} {1, 2, 1}         {0, 1,      |
|      -----------------------------------------------------------------------|
|                             2                                               |
|        p         p         p          - 15823p         p         p          |
|      0} {1, 0, 0} {1, 1, 1} {1, 2, 1}         {0, 1, 1} {1, 0, 0} {1, 1,    |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p          - 7631p         p         p         p          - 5446p    |
|      1} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                2                 2                   2      |
|           p         p         p          - 14756p         p         p       |
|      2, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1}         {1, 0, 0} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|         - 7544p         p         p         p          - 8467p         p    |
|      1}        {0, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p          - 11384p         p         p         p       |
|      0, 1} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, 0, 1} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|         + 3701p         p         p         p          - 1954p         p    |
|      1}        {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {1, |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          + 9427p         p         p         p        |
|      0, 1} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 0, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                               2                2            |
|         + 15183p         p         p         p          - 6468p         p   |
|      1}         {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p          + 7891p         p         p         p          -       |
|      1, 1} {1, 2, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}         |
|      -----------------------------------------------------------------------|
|                                         2                                   |
|      1679p         p         p         p          - 8532p         p         |
|           {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {1, 1,   |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p          + 2900p         p         p         p          +|
|      0} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      13937p         p         p         p          - 4999p         p        |
|            {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                   2                2                   2                    |
|        p         p          - 2921p         p         p          - 15042p   |
|      0} {1, 1, 1} {1, 2, 1}        {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2         2                           2         2                |
|           p         p          + 11165p         p         p          -      |
|      0, 1} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, 1, 1} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                      2         2                          2         2       |
|      14266p         p         p          - 1034p         p         p        |
|            {0, 1, 1} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                          2         2                           2         2  |
|         + 7716p         p         p          + 13876p         p         p   |
|      1}        {0, 2, 1} {1, 1, 1} {1, 2, 1}         {1, 0, 0} {1, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|                            2         2                          2           |
|            - 537p         p         p          + 1647p         p            |
|      2, 1}       {1, 0, 1} {1, 1, 1} {1, 2, 1}        {1, 1, 0} {1, 1,      |
|      -----------------------------------------------------------------------|
|         2                 3         2                                       |
|        p          + 14968p         p          - 3573p         p         p   |
|      1} {1, 2, 1}         {1, 1, 1} {1, 2, 1}        {0, 0, 1} {1, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p          - 7147p         p         p         p          +       |
|      2, 0} {1, 2, 1}        {0, 1, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}         |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      10802p         p         p         p          + 13653p         p       |
|            {0, 2, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}         {1, 0, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p          - 8888p         p         p         p          +|
|      0} {1, 2, 0} {1, 2, 1}        {1, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|           2                   2                                             |
|      7437p         p         p          - 15336p         p         p        |
|           {1, 1, 0} {1, 2, 0} {1, 2, 1}         {0, 0, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p          - 1097p         p         p         p          - 2112p    |
|      0} {1, 2, 1}        {0, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          - 3090p         p         p        |
|      1, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 2, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p          + 10072p         p         p         p          + 4311p   |
|      0} {1, 2, 1}         {1, 0, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {1,|
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          - 10100p         p         p       |
|      0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {1, 1, 0} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|         2               2                   2                           2   |
|        p          - 675p         p         p          - 10993p         p    |
|      0} {1, 2, 1}       {1, 1, 1} {1, 2, 0} {1, 2, 1}         {1, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|            2                           2         2                          |
|           p          + 11604p         p         p          - 3130p          |
|      2, 0} {1, 2, 1}         {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 0,    |
|      -----------------------------------------------------------------------|
|                   3                                    3                    |
|        p         p          - 6085p         p         p          + 2595p    |
|      1} {1, 1, 0} {1, 2, 1}        {0, 1, 0} {1, 1, 0} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                      3                                     3                |
|           p         p          + 11817p         p         p          +      |
|      2, 0} {1, 1, 0} {1, 2, 1}         {0, 2, 1} {1, 1, 0} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                               3                                    3        |
|      9509p         p         p          - 5177p         p         p         |
|           {1, 0, 0} {1, 1, 0} {1, 2, 1}        {1, 0, 1} {1, 1, 0} {1, 2, 1}|
|      -----------------------------------------------------------------------|
|             2         3                                     3               |
|      - 8772p         p          + 15994p         p         p          -     |
|             {1, 1, 0} {1, 2, 1}         {0, 0, 1} {1, 1, 1} {1, 2, 1}       |
|      -----------------------------------------------------------------------|
|                             3                                    3          |
|      87p         p         p          - 7066p         p         p          +|
|         {0, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                3                                    3       |
|      11128p         p         p          + 9739p         p         p        |
|            {0, 2, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                     3                                       |
|         - 11542p         p         p          + 15725p         p            |
|      1}         {1, 0, 0} {1, 1, 1} {1, 2, 1}         {1, 0, 1} {1, 1,      |
|      -----------------------------------------------------------------------|
|         3                                    3                2         3   |
|        p          - 6632p         p         p          - 8538p         p    |
|      1} {1, 2, 1}        {1, 1, 0} {1, 1, 1} {1, 2, 1}        {1, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|                                       3                                     |
|            - 4007p         p         p          - 3055p         p           |
|      2, 1}        {1, 1, 0} {1, 2, 0} {1, 2, 1}        {1, 1, 1} {1, 2,     |
|      -----------------------------------------------------------------------|
|         3           2                                                       |
|        p         , p         p         p         p          + 613p          |
|      0} {1, 2, 1}   {0, 0, 1} {0, 1, 1} {0, 2, 1} {1, 0, 1}       {0, 0,    |
|      -----------------------------------------------------------------------|
|                                                                  2          |
|        p         p         p         p          - 1591p         p           |
|      1} {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 0, 1}        {0, 0, 0} {0, 1,     |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 6768p         p         p         p          +|
|      1} {0, 2, 1} {1, 0, 1}        {0, 0, 1} {0, 1, 1} {0, 2, 1} {1, 0, 1}  |
|      -----------------------------------------------------------------------|
|                      2                                    2                 |
|      14622p         p         p         p          + 5876p         p        |
|            {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 0, 1}        {0, 1, 1} {0, 2,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 1314p         p         p         p          -|
|      0} {0, 2, 1} {1, 0, 1}        {0, 0, 1} {0, 2, 0} {0, 2, 1} {1, 0, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                    2         2        |
|      9198p         p         p         p          - 8631p         p         |
|           {0, 1, 1} {0, 2, 0} {0, 2, 1} {1, 0, 1}        {0, 0, 1} {0, 2,   |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p          + 12466p         p         p         p          - 879p    |
|      1} {1, 0, 1}         {0, 0, 1} {0, 1, 0} {0, 2, 1} {1, 0, 1}       {0, |
|      -----------------------------------------------------------------------|
|                      2                                             2        |
|           p         p         p          + 381p         p         p         |
|      0, 0} {0, 1, 1} {0, 2, 1} {1, 0, 1}       {0, 0, 1} {0, 1, 1} {0, 2,   |
|      -----------------------------------------------------------------------|
|                                               2                          2  |
|        p          + 12697p         p         p         p          - 4919p   |
|      1} {1, 0, 1}         {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 0, 1}        {0,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          - 8881p         p         p         p        |
|      1, 1} {0, 2, 1} {1, 0, 1}        {0, 0, 1} {0, 2, 0} {0, 2, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                    2                           2         2  |
|         + 6913p         p         p         p          + 11568p         p   |
|      1}        {0, 1, 1} {0, 2, 0} {0, 2, 1} {1, 0, 1}         {0, 2, 0} {0,|
|      -----------------------------------------------------------------------|
|                                       3                                     |
|           p          - 7675p         p         p          - 1777p           |
|      2, 1} {1, 0, 1}        {0, 0, 0} {0, 2, 1} {1, 0, 1}        {0, 0,     |
|      -----------------------------------------------------------------------|
|         3                                    3                              |
|        p         p          - 3010p         p         p          - 5676p    |
|      1} {0, 2, 1} {1, 0, 1}        {0, 1, 0} {0, 2, 1} {1, 0, 1}        {0, |
|      -----------------------------------------------------------------------|
|            3                                     3                          |
|           p         p          - 10528p         p         p          +      |
|      1, 1} {0, 2, 1} {1, 0, 1}         {0, 2, 0} {0, 2, 1} {1, 0, 1}        |
|      -----------------------------------------------------------------------|
|           2                                               2                 |
|      3022p         p         p         p          + 13196p         p        |
|           {0, 1, 1} {0, 2, 1} {1, 0, 0} {1, 0, 1}         {0, 2, 0} {0, 2,  |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          + 14066p         p         p         p          |
|      1} {1, 0, 0} {1, 0, 1}         {0, 1, 0} {0, 2, 1} {1, 0, 0} {1, 0, 1} |
|      -----------------------------------------------------------------------|
|                       2                                               2     |
|      + 1584p         p         p         p          + 15360p         p      |
|             {0, 1, 1} {0, 2, 1} {1, 0, 0} {1, 0, 1}         {0, 2, 0} {0, 2,|
|      -----------------------------------------------------------------------|
|                                     3                                       |
|        p         p          - 15681p         p         p          - 8262p   |
|      1} {1, 0, 0} {1, 0, 1}         {0, 2, 1} {1, 0, 0} {1, 0, 1}        {0,|
|      -----------------------------------------------------------------------|
|                      2                          2         2                 |
|           p         p         p          + 2251p         p         p        |
|      2, 0} {0, 2, 1} {1, 0, 0} {1, 0, 1}        {0, 2, 1} {1, 0, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|         - 6142p         p         p         p          - 12533p         p   |
|      1}        {0, 0, 0} {0, 1, 1} {0, 2, 1} {1, 0, 1}         {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p          + 14086p         p         p         p       |
|      1, 1} {0, 2, 1} {1, 0, 1}         {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 0, |
|      -----------------------------------------------------------------------|
|                 2                   2                                       |
|         + 13723p         p         p          - 7043p         p         p   |
|      1}         {0, 1, 1} {0, 2, 1} {1, 0, 1}        {0, 0, 1} {0, 2, 0} {0,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p          + 6184p         p         p         p          +       |
|      2, 1} {1, 0, 1}        {0, 1, 1} {0, 2, 0} {0, 2, 1} {1, 0, 1}         |
|      -----------------------------------------------------------------------|
|           2                   2                          2         2        |
|      9851p         p         p          - 1383p         p         p         |
|           {0, 2, 0} {0, 2, 1} {1, 0, 1}        {0, 0, 0} {0, 2, 1} {1, 0, 1}|
|      -----------------------------------------------------------------------|
|                       2         2                         2         2       |
|      + 4912p         p         p          + 295p         p         p        |
|             {0, 0, 1} {0, 2, 1} {1, 0, 1}       {0, 1, 0} {0, 2, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                          2         2                          2         2   |
|         - 4295p         p         p          + 4629p         p         p    |
|      1}        {0, 1, 1} {0, 2, 1} {1, 0, 1}        {0, 2, 0} {0, 2, 1} {1, |
|      -----------------------------------------------------------------------|
|                    3         2                                              |
|            - 11005p         p          - 11620p         p         p         |
|      0, 1}         {0, 2, 1} {1, 0, 1}         {0, 1, 1} {0, 2, 1} {1, 0,   |
|      -----------------------------------------------------------------------|
|         2                                              2                 2  |
|        p          + 9337p         p         p         p          + 14007p   |
|      0} {1, 0, 1}        {0, 2, 0} {0, 2, 1} {1, 0, 0} {1, 0, 1}         {0,|
|      -----------------------------------------------------------------------|
|                      2                                    3                 |
|           p         p          + 1175p         p         p          -       |
|      2, 1} {1, 0, 0} {1, 0, 1}        {0, 2, 0} {0, 2, 1} {1, 0, 1}         |
|      -----------------------------------------------------------------------|
|            2         3                          2                           |
|      14702p         p          + 6279p         p         p         p        |
|            {0, 2, 1} {1, 0, 1}        {0, 0, 1} {0, 2, 0} {0, 2, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                 2         2                                                 |
|         + 11694p         p         p          + 12774p         p            |
|      0}         {0, 0, 1} {0, 2, 1} {1, 1, 0}         {0, 0, 1} {0, 1,      |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p         p          + 5956p         p         p         p          +|
|      0} {0, 2, 1} {1, 1, 0}        {0, 0, 1} {0, 2, 0} {0, 2, 1} {1, 1, 0}  |
|      -----------------------------------------------------------------------|
|          2         2                                     3                  |
|      986p         p         p          + 12523p         p         p         |
|          {0, 2, 0} {0, 2, 1} {1, 1, 0}         {0, 0, 1} {0, 2, 1} {1, 1, 0}|
|      -----------------------------------------------------------------------|
|                        3                                    3               |
|      - 13648p         p         p          + 2384p         p         p      |
|              {0, 1, 0} {0, 2, 1} {1, 1, 0}        {0, 2, 0} {0, 2, 1} {1, 1,|
|      -----------------------------------------------------------------------|
|                                     2                           2           |
|         + 10270p         p         p         p          - 13076p            |
|      0}         {0, 2, 0} {0, 2, 1} {1, 0, 0} {1, 1, 0}         {0, 2,      |
|      -----------------------------------------------------------------------|
|         2                                     3                          2  |
|        p         p          + 14402p         p         p          - 8312p   |
|      1} {1, 0, 0} {1, 1, 0}         {0, 2, 1} {1, 0, 0} {1, 1, 0}        {0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p          + 4016p         p         p        |
|      0, 1} {0, 2, 1} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {0, 1, 0} {0, 2,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          - 2610p         p         p         p          +|
|      1} {1, 0, 1} {1, 1, 0}        {0, 1, 0} {0, 2, 1} {1, 0, 1} {1, 1, 0}  |
|      -----------------------------------------------------------------------|
|                                                                     2       |
|      15364p         p         p         p         p          - 8029p        |
|            {0, 0, 1} {0, 2, 0} {0, 2, 1} {1, 0, 1} {1, 1, 0}        {0, 2,  |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p          - 3670p         p         p           |
|      0} {0, 2, 1} {1, 0, 1} {1, 1, 0}        {0, 0, 0} {0, 2, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          + 12738p         p         p         p          + 3360p   |
|      1} {1, 1, 0}         {0, 0, 1} {0, 2, 1} {1, 0, 1} {1, 1, 0}        {0,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p         p          + 12180p         p         p       |
|      1, 0} {0, 2, 1} {1, 0, 1} {1, 1, 0}         {0, 2, 0} {0, 2, 1} {1, 0, |
|      -----------------------------------------------------------------------|
|                           3                                                 |
|        p          - 13563p         p         p          - 9023p         p   |
|      1} {1, 1, 0}         {0, 2, 1} {1, 0, 1} {1, 1, 0}        {0, 2, 0} {0,|
|      -----------------------------------------------------------------------|
|                                                  2                          |
|           p         p         p          - 10706p         p         p       |
|      2, 1} {1, 0, 0} {1, 0, 1} {1, 1, 0}         {0, 2, 1} {1, 0, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p          + 6161p         p         p         p          + 751p     |
|      1} {1, 1, 0}        {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1, 0}       {0,  |
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p         p          + 2358p         p         p        |
|      2, 0} {1, 0, 0} {1, 0, 1} {1, 1, 0}        {0, 2, 1} {1, 0, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                          3                                   2         2    |
|        p          - 7180p         p         p          - 782p         p     |
|      1} {1, 1, 0}        {1, 0, 0} {1, 0, 1} {1, 1, 0}       {0, 0, 1} {1,  |
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p          + 5450p         p         p         p          -       |
|      0, 1} {1, 1, 0}        {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1, 0}         |
|      -----------------------------------------------------------------------|
|           2         2                                              2        |
|      5998p         p         p          + 8831p         p         p         |
|           {0, 1, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {0, 2, 0} {1, 0,   |
|      -----------------------------------------------------------------------|
|                          2         2                                        |
|        p          - 9215p         p         p          + 2845p         p    |
|      1} {1, 1, 0}        {0, 2, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 0} {0, |
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          + 5214p         p         p         p        |
|      2, 1} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {0, 2, 1} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         - 8456p         p         p         p          - 1895p         p    |
|      0}        {0, 1, 0} {0, 2, 1} {1, 0, 1} {1, 1, 0}        {0, 2, 0} {0, |
|      -----------------------------------------------------------------------|
|            2                           2         2                          |
|           p         p          - 11933p         p         p          +      |
|      2, 1} {1, 0, 1} {1, 1, 0}         {0, 2, 1} {1, 0, 1} {1, 1, 0}        |
|      -----------------------------------------------------------------------|
|                               2                                             |
|      8238p         p         p         p          + 12998p         p        |
|           {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1, 0}         {0, 2, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p         p          - 10317p         p         p         p          |
|      0} {1, 0, 1} {1, 1, 0}         {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 1, 0} |
|      -----------------------------------------------------------------------|
|              2         2                                    3               |
|      - 14286p         p         p          - 5666p         p         p      |
|              {1, 0, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {1, 0, 1} {1, 1,|
|      -----------------------------------------------------------------------|
|                          3                                     3            |
|         + 2011p         p         p          + 11891p         p         p   |
|      0}        {0, 1, 0} {1, 0, 1} {1, 1, 0}         {0, 2, 0} {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                            3                                    3           |
|            - 881p         p         p          - 2222p         p            |
|      1, 0}       {0, 2, 1} {1, 0, 1} {1, 1, 0}        {1, 0, 0} {1, 0,      |
|      -----------------------------------------------------------------------|
|                          2                   2                              |
|        p          + 3909p         p         p          - 11378p         p   |
|      1} {1, 1, 0}        {0, 0, 1} {0, 2, 1} {1, 1, 0}         {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|                      2                 2                   2                |
|           p         p          + 10467p         p         p          -      |
|      1, 0} {0, 2, 1} {1, 1, 0}         {0, 1, 0} {0, 2, 1} {1, 1, 0}        |
|      -----------------------------------------------------------------------|
|                                          2                 2                |
|      15602p         p         p         p          + 15812p         p       |
|            {0, 0, 1} {0, 2, 0} {0, 2, 1} {1, 1, 0}         {0, 2, 0} {0, 2, |
|      -----------------------------------------------------------------------|
|         2                           2         2                          2  |
|        p          + 13060p         p         p          - 6498p         p   |
|      1} {1, 1, 0}         {0, 0, 1} {0, 2, 1} {1, 1, 0}        {0, 1, 0} {0,|
|      -----------------------------------------------------------------------|
|            2                           2         2                3         |
|           p          + 12548p         p         p          + 9792p          |
|      2, 1} {1, 1, 0}         {0, 2, 0} {0, 2, 1} {1, 1, 0}        {0, 2,    |
|      -----------------------------------------------------------------------|
|         2                                              2                2   |
|        p          + 4645p         p         p         p          + 2848p    |
|      1} {1, 1, 0}        {0, 2, 0} {0, 2, 1} {1, 0, 0} {1, 1, 0}        {0, |
|      -----------------------------------------------------------------------|
|                      2                           2         2                |
|           p         p          + 10023p         p         p          -      |
|      2, 1} {1, 0, 0} {1, 1, 0}         {0, 2, 1} {1, 0, 0} {1, 1, 0}        |
|      -----------------------------------------------------------------------|
|          2                   2                                              |
|      683p         p         p          - 1741p         p         p          |
|          {0, 0, 1} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {0, 1, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|         2                2                   2                              |
|        p          - 4138p         p         p          - 3779p         p    |
|      1} {1, 1, 0}        {0, 1, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {0, |
|      -----------------------------------------------------------------------|
|                      2                 2                   2                |
|           p         p          - 15224p         p         p          +      |
|      2, 0} {1, 0, 1} {1, 1, 0}         {0, 2, 0} {1, 0, 1} {1, 1, 0}        |
|      -----------------------------------------------------------------------|
|                                         2                                   |
|      3620p         p         p         p          + 5198p         p         |
|           {0, 0, 0} {0, 2, 1} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {0, 2,   |
|      -----------------------------------------------------------------------|
|                   2                                               2         |
|        p         p          - 11865p         p         p         p          |
|      1} {1, 0, 1} {1, 1, 0}         {0, 1, 0} {0, 2, 1} {1, 0, 1} {1, 1, 0} |
|      -----------------------------------------------------------------------|
|                                            2                2               |
|      + 14792p         p         p         p          + 1764p         p      |
|              {0, 2, 0} {0, 2, 1} {1, 0, 1} {1, 1, 0}        {0, 2, 1} {1, 0,|
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p          + 7357p         p         p         p          + 2582p    |
|      1} {1, 1, 0}        {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1, 0}        {0, |
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          + 498p         p         p         |
|      2, 0} {1, 0, 0} {1, 0, 1} {1, 1, 0}       {0, 2, 1} {1, 0, 0} {1, 0,   |
|      -----------------------------------------------------------------------|
|         2                 2                   2                             |
|        p          + 12014p         p         p          + 11366p            |
|      1} {1, 1, 0}         {1, 0, 0} {1, 0, 1} {1, 1, 0}         {0, 0,      |
|      -----------------------------------------------------------------------|
|         2         2                           2         2                   |
|        p         p          + 12242p         p         p          +         |
|      0} {1, 0, 1} {1, 1, 0}         {0, 0, 1} {1, 0, 1} {1, 1, 0}           |
|      -----------------------------------------------------------------------|
|                      2         2                           2         2      |
|      15987p         p         p          - 12309p         p         p       |
|            {0, 1, 0} {1, 0, 1} {1, 1, 0}         {0, 2, 0} {1, 0, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|                           2         2                          2         2  |
|         - 15440p         p         p          + 1242p         p         p   |
|      0}         {0, 2, 1} {1, 0, 1} {1, 1, 0}        {1, 0, 0} {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                   3         2                2         3                    |
|            - 9919p         p          - 1974p         p          + 471p     |
|      1, 0}        {1, 0, 1} {1, 1, 0}        {0, 0, 1} {1, 1, 0}       {0,  |
|      -----------------------------------------------------------------------|
|                      3                                     3                |
|           p         p          - 12114p         p         p          +      |
|      0, 1} {0, 1, 0} {1, 1, 0}         {0, 0, 1} {0, 2, 0} {1, 1, 0}        |
|      -----------------------------------------------------------------------|
|                                3                                    3       |
|      10941p         p         p          - 7958p         p         p        |
|            {0, 0, 1} {0, 2, 1} {1, 1, 0}        {0, 1, 0} {0, 2, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                    3                2         3             |
|         + 6871p         p         p          + 2717p         p          -   |
|      0}        {0, 2, 0} {0, 2, 1} {1, 1, 0}        {0, 2, 1} {1, 1, 0}     |
|      -----------------------------------------------------------------------|
|                               3                                    3        |
|      8736p         p         p          - 2676p         p         p         |
|           {0, 2, 1} {1, 0, 0} {1, 1, 0}        {0, 0, 0} {1, 0, 1} {1, 1, 0}|
|      -----------------------------------------------------------------------|
|                                 3                                     3     |
|      + 1206p         p         p          - 15461p         p         p      |
|             {0, 0, 1} {1, 0, 1} {1, 1, 0}         {0, 1, 0} {1, 0, 1} {1, 1,|
|      -----------------------------------------------------------------------|
|                                    3                                     3  |
|         + 6056p         p         p          - 11532p         p         p   |
|      0}        {0, 2, 0} {1, 0, 1} {1, 1, 0}         {0, 2, 1} {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                                      3                 2         3          |
|            + 596p         p         p          - 14501p         p          +|
|      1, 0}       {1, 0, 0} {1, 0, 1} {1, 1, 0}         {1, 0, 1} {1, 1, 0}  |
|      -----------------------------------------------------------------------|
|                     4                          4                            |
|      3968p         p          - 8684p         p          + 7059p            |
|           {0, 0, 1} {1, 1, 0}        {0, 2, 1} {1, 1, 0}        {1, 0,      |
|      -----------------------------------------------------------------------|
|         4                           2                                    2  |
|        p          - 11258p         p         p         p          + 4411p   |
|      1} {1, 1, 0}         {0, 0, 1} {0, 1, 0} {0, 2, 1} {1, 1, 1}        {0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p          + 10296p         p         p       |
|      0, 1} {0, 1, 1} {0, 2, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 0} {0, 1, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          + 2066p         p         p         p          -|
|      1} {0, 2, 1} {1, 1, 1}        {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                              2        |
|      8468p         p         p         p          + 2506p         p         |
|           {0, 0, 1} {0, 1, 1} {0, 2, 1} {1, 1, 1}        {0, 1, 0} {0, 1,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 14161p         p         p         p          |
|      1} {0, 2, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 1} {0, 2, 0} {0, 2,    |
|      -----------------------------------------------------------------------|
|                           2                                                 |
|        p          + 14823p         p         p         p          - 7737p   |
|      1} {1, 1, 1}         {0, 1, 1} {0, 2, 0} {0, 2, 1} {1, 1, 1}        {0,|
|      -----------------------------------------------------------------------|
|            2                                             2                  |
|           p         p         p          + 311p         p         p         |
|      0, 1} {0, 2, 0} {0, 2, 1} {1, 1, 1}       {0, 1, 1} {0, 2, 0} {0, 2,   |
|      -----------------------------------------------------------------------|
|                          2         2                                        |
|        p          + 6838p         p         p          - 12834p         p   |
|      1} {1, 1, 1}        {0, 0, 1} {0, 2, 1} {1, 1, 1}         {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|            2                           2         2                          |
|           p         p          - 13762p         p         p          +      |
|      1, 0} {0, 2, 1} {1, 1, 1}         {0, 1, 0} {0, 2, 1} {1, 1, 1}        |
|      -----------------------------------------------------------------------|
|                               2                                             |
|      2566p         p         p         p          + 5359p         p         |
|           {0, 0, 1} {0, 1, 1} {0, 2, 1} {1, 1, 1}        {0, 1, 0} {0, 1,   |
|      -----------------------------------------------------------------------|
|         2                           2         2                             |
|        p         p          - 14519p         p         p          - 3242p   |
|      1} {0, 2, 1} {1, 1, 1}         {0, 1, 1} {0, 2, 1} {1, 1, 1}        {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p         p          + 7421p         p         p        |
|      0, 1} {0, 2, 0} {0, 2, 1} {1, 1, 1}        {0, 1, 1} {0, 2, 0} {0, 2,  |
|      -----------------------------------------------------------------------|
|                          2         2                                     3  |
|        p          - 5978p         p         p          - 15811p         p   |
|      1} {1, 1, 1}        {0, 2, 0} {0, 2, 1} {1, 1, 1}         {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|                                        3                                    |
|           p          + 15948p         p         p          - 7610p          |
|      2, 1} {1, 1, 1}         {0, 1, 0} {0, 2, 1} {1, 1, 1}        {0, 1,    |
|      -----------------------------------------------------------------------|
|         3                                    3                           2  |
|        p         p          - 8287p         p         p          + 10197p   |
|      1} {0, 2, 1} {1, 1, 1}        {0, 2, 0} {0, 2, 1} {1, 1, 1}         {0,|
|      -----------------------------------------------------------------------|
|                                                  2                          |
|           p         p         p          + 11550p         p         p       |
|      1, 1} {0, 2, 1} {1, 0, 0} {1, 1, 1}         {0, 2, 0} {0, 2, 1} {1, 0, |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          + 15158p         p         p         p          + 2777p   |
|      0} {1, 1, 1}         {0, 1, 0} {0, 2, 1} {1, 0, 0} {1, 1, 1}        {0,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p         p          + 7776p         p         p        |
|      1, 1} {0, 2, 1} {1, 0, 0} {1, 1, 1}        {0, 2, 0} {0, 2, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                          3                                     2         2  |
|        p          - 1983p         p         p          - 13931p         p   |
|      0} {1, 1, 1}        {0, 2, 1} {1, 0, 0} {1, 1, 1}         {0, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|                              2         2                                    |
|           p          - 13489p         p         p          - 15162p         |
|      0, 0} {1, 1, 1}         {0, 2, 0} {1, 0, 0} {1, 1, 1}         {0, 1,   |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p         p          - 2341p         p         p           |
|      0} {0, 2, 1} {1, 0, 0} {1, 1, 1}        {0, 1, 1} {0, 2, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                               2                          2  |
|        p          + 14466p         p         p         p          + 7137p   |
|      0} {1, 1, 1}         {0, 2, 0} {0, 2, 1} {1, 0, 0} {1, 1, 1}        {0,|
|      -----------------------------------------------------------------------|
|            2                                    3                           |
|           p         p          + 4038p         p         p          +       |
|      2, 1} {1, 0, 0} {1, 1, 1}        {0, 1, 0} {1, 0, 0} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|                     3                                    3                  |
|      1226p         p         p          - 5742p         p         p         |
|           {0, 1, 1} {1, 0, 0} {1, 1, 1}        {0, 2, 0} {1, 0, 0} {1, 1, 1}|
|      -----------------------------------------------------------------------|
|                        3                           3                        |
|      + 14146p         p         p          - 12580p         p         p     |
|              {0, 2, 1} {1, 0, 0} {1, 1, 1}         {0, 1, 0} {1, 0, 1} {1,  |
|      -----------------------------------------------------------------------|
|                    2                                                        |
|            - 15106p         p         p         p          + 15618p         |
|      1, 1}         {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 1, 1}         {0, 0,   |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p         p          + 5941p         p           |
|      1} {0, 1, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {0, 1,     |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          - 10121p         p         p         p          |
|      1} {1, 0, 1} {1, 1, 1}         {0, 0, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|                       2                                               2     |
|      - 2156p         p         p         p          + 11068p         p      |
|             {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 1, 1}         {0, 1, 0} {0, 1,|
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p         p          - 15861p         p         p         p          |
|      1} {1, 0, 1} {1, 1, 1}         {0, 1, 1} {0, 2, 0} {1, 0, 1} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|                       2                                               2     |
|      + 4724p         p         p         p          - 13093p         p      |
|             {0, 0, 1} {0, 2, 0} {1, 0, 1} {1, 1, 1}         {0, 1, 1} {0, 2,|
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p         p          - 11350p         p         p         p          |
|      0} {1, 0, 1} {1, 1, 1}         {0, 0, 1} {0, 2, 1} {1, 0, 1} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|                                                                       2     |
|      + 14162p         p         p         p         p          + 3306p      |
|              {0, 0, 1} {0, 1, 0} {0, 2, 1} {1, 0, 1} {1, 1, 1}        {0, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 1953p         p         p           |
|      0} {0, 2, 1} {1, 0, 1} {1, 1, 1}        {0, 0, 0} {0, 1, 1} {0, 2,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 15636p         p         p         p          |
|      1} {1, 0, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 1} {0, 2, 1} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 1934p         p         p         p         p          +|
|      1} {1, 1, 1}        {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 0, 1} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|           2                                                                 |
|      1669p         p         p         p          + 12716p         p        |
|           {0, 1, 1} {0, 2, 1} {1, 0, 1} {1, 1, 1}         {0, 0, 1} {0, 2,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 867p         p         p            |
|      0} {0, 2, 1} {1, 0, 1} {1, 1, 1}       {0, 1, 1} {0, 2, 0} {0, 2,      |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          - 3848p         p         p         p          -|
|      1} {1, 0, 1} {1, 1, 1}        {0, 2, 0} {0, 2, 1} {1, 0, 1} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                              2        |
|      6935p         p         p         p          - 3703p         p         |
|           {0, 0, 0} {0, 2, 1} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {0, 2,   |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 2818p         p         p         p          -|
|      1} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {0, 2, 1} {1, 0, 1} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                              2        |
|      1805p         p         p         p          - 7267p         p         |
|           {0, 1, 1} {0, 2, 1} {1, 0, 1} {1, 1, 1}        {0, 2, 0} {0, 2,   |
|      -----------------------------------------------------------------------|
|                                    3                                    2   |
|        p         p          - 5170p         p         p          - 3742p    |
|      1} {1, 0, 1} {1, 1, 1}        {0, 2, 1} {1, 0, 1} {1, 1, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p         p         p          - 5465p         p         p        |
|      1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}        {0, 2, 0} {1, 0, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 5174p         p         p         p         p          +|
|      1} {1, 1, 1}        {0, 1, 0} {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      3382p         p         p         p         p          + 14908p        |
|           {0, 1, 1} {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}         {0, 2,  |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p         p          - 9727p         p           |
|      0} {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}        {0, 2, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          - 15266p         p         p         p          |
|      0} {1, 0, 1} {1, 1, 1}         {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|                       2                                               2     |
|      - 2643p         p         p         p          + 13888p         p      |
|             {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}         {0, 2, 0} {1, 0,|
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          + 6641p         p         p         p          -|
|      0} {1, 0, 1} {1, 1, 1}        {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|            3                                    2         2                 |
|      10069p         p         p          - 7919p         p         p        |
|            {1, 0, 0} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                    2                          2         2   |
|         - 8824p         p         p         p          + 9385p         p    |
|      1}        {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p          - 1749p         p         p         p          +       |
|      0, 1} {1, 1, 1}        {0, 0, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|                               2                                             |
|      7904p         p         p         p          - 14598p         p        |
|           {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 1, 1}         {0, 1, 0} {0, 1,  |
|      -----------------------------------------------------------------------|
|         2                          2         2                              |
|        p         p          + 1828p         p         p          + 12770p   |
|      1} {1, 0, 1} {1, 1, 1}        {0, 1, 1} {1, 0, 1} {1, 1, 1}         {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p         p          - 3083p         p         p        |
|      0, 1} {0, 2, 0} {1, 0, 1} {1, 1, 1}        {0, 1, 1} {0, 2, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                          2         2                                        |
|        p          + 9257p         p         p          - 9268p         p    |
|      1} {1, 1, 1}        {0, 2, 0} {1, 0, 1} {1, 1, 1}        {0, 0, 0} {0, |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          + 10651p         p         p         p       |
|      2, 1} {1, 0, 1} {1, 1, 1}         {0, 0, 1} {0, 2, 1} {1, 0, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         - 4679p         p         p         p          - 2682p         p    |
|      1}        {0, 1, 0} {0, 2, 1} {1, 0, 1} {1, 1, 1}        {0, 1, 1} {0, |
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          + 5278p         p         p         p        |
|      2, 1} {1, 0, 1} {1, 1, 1}        {0, 2, 0} {0, 2, 1} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                2         2                                               2  |
|         - 8994p         p         p          + 10406p         p         p   |
|      1}        {0, 2, 1} {1, 0, 1} {1, 1, 1}         {0, 1, 0} {1, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|                                                  2                          |
|           p          - 13141p         p         p         p          +      |
|      0, 1} {1, 1, 1}         {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}        |
|      -----------------------------------------------------------------------|
|                               2                                             |
|      6919p         p         p         p          + 12835p         p        |
|           {0, 2, 0} {1, 0, 0} {1, 0, 1} {1, 1, 1}         {0, 2, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|         2                         2         2                               |
|        p         p          + 796p         p         p          + 15141p    |
|      0} {1, 0, 1} {1, 1, 1}       {1, 0, 0} {1, 0, 1} {1, 1, 1}         {0, |
|      -----------------------------------------------------------------------|
|            3                                    3                           |
|           p         p          + 9356p         p         p          +       |
|      0, 1} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|                      3                                    3                 |
|      12551p         p         p          - 6056p         p         p        |
|            {0, 1, 1} {1, 0, 1} {1, 1, 1}        {0, 2, 0} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                          3                                     3            |
|         - 1524p         p         p          - 12195p         p         p   |
|      1}        {0, 2, 1} {1, 0, 1} {1, 1, 1}         {1, 0, 0} {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                              2                                    3         |
|            - 15193p         p         p         p          - 5831p          |
|      1, 1}         {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 1, 1}        {0, 1,    |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 4015p         p         p         p          +|
|      0} {1, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 2, 0} {1, 1, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|            2                                                                |
|      11634p         p         p         p          + 10077p         p       |
|            {0, 0, 1} {0, 2, 1} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {0, 1, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p         p          - 5022p         p         p           |
|      0} {0, 2, 1} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {0, 2, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 13516p         p         p         p         p          |
|      0} {1, 1, 1}         {0, 0, 1} {0, 2, 0} {0, 2, 1} {1, 1, 0} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|              2                                                        2     |
|      - 10019p         p         p         p          - 4091p         p      |
|              {0, 2, 0} {0, 2, 1} {1, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 2,|
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 9910p         p         p         p          -|
|      1} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {0, 2, 1} {1, 1, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                   3                   |
|      6633p         p         p         p          + 292p         p          |
|           {0, 2, 0} {0, 2, 1} {1, 1, 0} {1, 1, 1}       {0, 2, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 9493p         p         p         p         p          +|
|      0} {1, 1, 1}        {0, 2, 0} {0, 2, 1} {1, 0, 0} {1, 1, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|           2                                                       2         |
|      7173p         p         p         p          + 675p         p          |
|           {0, 2, 1} {1, 0, 0} {1, 1, 0} {1, 1, 1}       {0, 1, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          + 14300p         p         p         p          |
|      0} {1, 1, 0} {1, 1, 1}         {0, 2, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|                        2                                    3               |
|      - 13331p         p         p         p          + 3207p         p      |
|              {0, 2, 1} {1, 0, 0} {1, 1, 0} {1, 1, 1}        {1, 0, 0} {1, 1,|
|      -----------------------------------------------------------------------|
|                          2                                                  |
|        p          - 6620p         p         p         p          - 13095p   |
|      0} {1, 1, 1}        {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 1, 1}         {0,|
|      -----------------------------------------------------------------------|
|                                                           2                 |
|           p         p         p         p          + 1290p         p        |
|      0, 1} {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 8833p         p         p         p           |
|      1} {1, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 2, 0} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                          2                                                  |
|        p          + 9539p         p         p         p          - 745p     |
|      0} {1, 1, 1}        {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}       {0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p         p          + 11658p         p       |
|      0, 0} {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {0, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 6535p         p         p           |
|      1} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {0, 2, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 14554p         p         p         p          |
|      1} {1, 1, 0} {1, 1, 1}         {0, 2, 0} {0, 2, 1} {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                          2                                                  |
|        p          + 5081p         p         p         p          + 6509p    |
|      0} {1, 1, 1}        {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p         p          - 13661p         p       |
|      1, 0} {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 2, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 551p         p         p            |
|      0} {1, 0, 1} {1, 1, 0} {1, 1, 1}       {0, 2, 1} {1, 0, 0} {1, 0,      |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p         p          - 12419p         p         p         p          |
|      1} {1, 1, 0} {1, 1, 1}         {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|                       2                                               2     |
|      - 2520p         p         p         p          - 12800p         p      |
|             {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {1, 0,|
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          + 5538p         p         p         p          +|
|      1} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                              2        |
|      4002p         p         p         p          + 8244p         p         |
|           {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 2, 1} {1, 0,   |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          + 3484p         p         p         p          -|
|      1} {1, 1, 0} {1, 1, 1}        {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|           3                                    2         2                  |
|      3261p         p         p          - 3935p         p         p         |
|           {1, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}|
|      -----------------------------------------------------------------------|
|                                 2                           2         2     |
|      - 3852p         p         p         p          + 13529p         p      |
|             {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 1, 1}         {0, 1, 0} {1, 1,|
|      -----------------------------------------------------------------------|
|                                              2                           2  |
|        p          + 5302p         p         p         p          + 14978p   |
|      0} {1, 1, 1}        {0, 0, 1} {0, 2, 0} {1, 1, 0} {1, 1, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          - 13042p         p         p         p       |
|      2, 0} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {0, 2, 1} {1, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         + 2247p         p         p         p          + 4523p         p    |
|      1}        {0, 1, 0} {0, 2, 1} {1, 1, 0} {1, 1, 1}        {0, 2, 0} {0, |
|      -----------------------------------------------------------------------|
|            2                          2         2                           |
|           p         p          + 4753p         p         p          +       |
|      2, 1} {1, 1, 0} {1, 1, 1}        {0, 2, 1} {1, 1, 0} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|                                2                                            |
|      11452p         p         p         p          - 1464p         p        |
|            {0, 1, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1}        {0, 2, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p         p          - 2586p         p         p         p          -|
|      0} {1, 1, 0} {1, 1, 1}        {0, 2, 1} {1, 0, 0} {1, 1, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|           2         2                                              2        |
|      7984p         p         p          + 7729p         p         p         |
|           {1, 0, 0} {1, 1, 0} {1, 1, 1}        {0, 0, 0} {1, 0, 1} {1, 1,   |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          - 1565p         p         p         p          + 1217p    |
|      0} {1, 1, 1}        {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {0, |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p         p          - 12941p         p         p       |
|      1, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 2, 0} {1, 0, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          - 7694p         p         p         p          + 7587p    |
|      0} {1, 1, 1}        {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {1, |
|      -----------------------------------------------------------------------|
|                      2                           2         2                |
|           p         p         p          - 10536p         p         p       |
|      0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                          3                                     3            |
|         + 3185p         p         p          - 13877p         p         p   |
|      1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 1, 0} {1, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|                              3                                     3        |
|            - 12569p         p         p          - 15271p         p         |
|      1, 1}         {0, 2, 0} {1, 1, 0} {1, 1, 1}         {0, 2, 1} {1, 1,   |
|      -----------------------------------------------------------------------|
|                                    3                                    3   |
|        p          + 3392p         p         p          - 6936p         p    |
|      0} {1, 1, 1}        {1, 0, 0} {1, 1, 0} {1, 1, 1}        {1, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|                             4                                     2         |
|           p          - 1466p         p          + 10585p         p          |
|      1, 0} {1, 1, 1}        {1, 1, 0} {1, 1, 1}         {0, 0, 1} {0, 1,    |
|      -----------------------------------------------------------------------|
|         2                3         2                2                   2   |
|        p          + 2787p         p          - 2320p         p         p    |
|      0} {1, 1, 1}        {0, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|                                                 2                2          |
|            + 7644p         p         p         p          + 4191p           |
|      1, 1}        {0, 0, 1} {0, 1, 0} {0, 1, 1} {1, 1, 1}        {0, 1,     |
|      -----------------------------------------------------------------------|
|                   2                           2         2                   |
|        p         p          - 12352p         p         p          -         |
|      0} {0, 1, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 1} {1, 1, 1}           |
|      -----------------------------------------------------------------------|
|                      2         2                                            |
|      12529p         p         p          + 11366p         p         p       |
|            {0, 1, 0} {0, 1, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 1} {0, 2, |
|      -----------------------------------------------------------------------|
|         2                2                   2                          2   |
|        p          - 3198p         p         p          - 6964p         p    |
|      0} {1, 1, 1}        {0, 1, 1} {0, 2, 0} {1, 1, 1}        {0, 0, 1} {0, |
|      -----------------------------------------------------------------------|
|            2                          2         2                2          |
|           p          + 8954p         p         p          + 7185p           |
|      2, 0} {1, 1, 1}        {0, 1, 1} {0, 2, 0} {1, 1, 1}        {0, 0,     |
|      -----------------------------------------------------------------------|
|                   2                                               2         |
|        p         p          - 10227p         p         p         p          |
|      1} {0, 2, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 0} {0, 2, 1} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|             2                   2                                           |
|      + 5883p         p         p          + 14814p         p         p      |
|             {0, 1, 0} {0, 2, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 1} {0, 2,|
|      -----------------------------------------------------------------------|
|         2                                              2                 2  |
|        p          + 8306p         p         p         p          + 10024p   |
|      1} {1, 1, 1}        {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 1, 1}         {0,|
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p          - 14436p         p         p         p       |
|      1, 1} {0, 2, 1} {1, 1, 1}         {0, 0, 1} {0, 2, 0} {0, 2, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|                                              2                 2            |
|         + 2857p         p         p         p          - 13713p         p   |
|      1}        {0, 1, 1} {0, 2, 0} {0, 2, 1} {1, 1, 1}         {0, 2, 0} {0,|
|      -----------------------------------------------------------------------|
|            2                           2         2                          |
|           p          + 12220p         p         p          + 2340p          |
|      2, 1} {1, 1, 1}         {0, 0, 1} {0, 2, 1} {1, 1, 1}        {0, 1,    |
|      -----------------------------------------------------------------------|
|         2         2                           2         2                   |
|        p         p          - 15353p         p         p          - p       |
|      0} {0, 2, 1} {1, 1, 1}         {0, 1, 1} {0, 2, 1} {1, 1, 1}    {0, 2, |
|      -----------------------------------------------------------------------|
|         2         2                 3         2                2            |
|        p         p          + 15658p         p          + 7288p         p   |
|      0} {0, 2, 1} {1, 1, 1}         {0, 2, 1} {1, 1, 1}        {0, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|            2                 2                   2                          |
|           p          - 12528p         p         p          - 14946p         |
|      0, 0} {1, 1, 1}         {0, 2, 0} {1, 0, 0} {1, 1, 1}         {0, 1,   |
|      -----------------------------------------------------------------------|
|                             2                                               |
|        p         p         p          - 710p         p         p            |
|      0} {0, 2, 1} {1, 0, 0} {1, 1, 1}       {0, 1, 1} {0, 2, 1} {1, 0,      |
|      -----------------------------------------------------------------------|
|         2                                               2               2   |
|        p          - 15130p         p         p         p          + 749p    |
|      0} {1, 1, 1}         {0, 2, 0} {0, 2, 1} {1, 0, 0} {1, 1, 1}       {0, |
|      -----------------------------------------------------------------------|
|                      2                          2         2                 |
|           p         p          + 1733p         p         p          + 89p   |
|      2, 1} {1, 0, 0} {1, 1, 1}        {0, 1, 0} {1, 0, 0} {1, 1, 1}      {0,|
|      -----------------------------------------------------------------------|
|            2         2                         2         2                  |
|           p         p          - 645p         p         p          +        |
|      1, 1} {1, 0, 0} {1, 1, 1}       {0, 2, 0} {1, 0, 0} {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|                      2         2                3         2                 |
|      10565p         p         p          - 5260p         p          +       |
|            {0, 2, 1} {1, 0, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|           2                   2                                             |
|      5443p         p         p          + 15816p         p         p        |
|           {0, 0, 1} {1, 0, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|         2               2                   2                               |
|        p          - 718p         p         p          - 13005p         p    |
|      1} {1, 1, 1}       {0, 1, 0} {1, 0, 1} {1, 1, 1}         {0, 0, 0} {0, |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          + 7562p         p         p         p        |
|      1, 1} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                              2                2             |
|         + 1951p         p         p         p          - 1620p         p    |
|      1}        {0, 1, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1}        {0, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|            2                                             2                  |
|           p          + 695p         p         p         p          +        |
|      0, 1} {1, 1, 1}       {0, 0, 1} {0, 2, 0} {1, 0, 1} {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|                                          2                2                 |
|      11468p         p         p         p          + 5704p         p        |
|            {0, 1, 1} {0, 2, 0} {1, 0, 1} {1, 1, 1}        {0, 2, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p          + 5143p         p         p         p          - 14303p   |
|      1} {1, 1, 1}        {0, 0, 0} {0, 2, 1} {1, 0, 1} {1, 1, 1}         {0,|
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          + 3140p         p         p        |
|      0, 1} {0, 2, 1} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {0, 2, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p          - 10246p         p         p         p          -         |
|      1} {1, 1, 1}         {0, 1, 1} {0, 2, 1} {1, 0, 1} {1, 1, 1}           |
|      -----------------------------------------------------------------------|
|                                          2                 2                |
|      13566p         p         p         p          - 12805p         p       |
|            {0, 2, 0} {0, 2, 1} {1, 0, 1} {1, 1, 1}         {0, 2, 1} {1, 0, |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p          - 13090p         p         p         p          -         |
|      1} {1, 1, 1}         {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1, 1}           |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      11365p         p         p         p          + 3270p         p        |
|            {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}        {0, 2, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                   2                                               2         |
|        p         p          + 14812p         p         p         p          |
|      0} {1, 0, 1} {1, 1, 1}         {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|              2                   2                          2         2     |
|      + 12555p         p         p          - 8070p         p         p      |
|              {1, 0, 0} {1, 0, 1} {1, 1, 1}        {0, 0, 0} {1, 0, 1} {1, 1,|
|      -----------------------------------------------------------------------|
|                           2         2                           2           |
|         - 14006p         p         p          - 13801p         p            |
|      1}         {0, 0, 1} {1, 0, 1} {1, 1, 1}         {0, 1, 0} {1, 0,      |
|      -----------------------------------------------------------------------|
|         2                           2         2                             |
|        p          + 13081p         p         p          + 14154p            |
|      1} {1, 1, 1}         {0, 1, 1} {1, 0, 1} {1, 1, 1}         {0, 2,      |
|      -----------------------------------------------------------------------|
|         2         2                          2         2                    |
|        p         p          + 2982p         p         p          + 2029p    |
|      0} {1, 0, 1} {1, 1, 1}        {0, 2, 1} {1, 0, 1} {1, 1, 1}        {1, |
|      -----------------------------------------------------------------------|
|            2         2                 3         2                2         |
|           p         p          - 10089p         p          - 8458p          |
|      0, 0} {1, 0, 1} {1, 1, 1}         {1, 0, 1} {1, 1, 1}        {0, 0,    |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p          - 3401p         p         p         p          +|
|      1} {1, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|           2                   2                                             |
|      1653p         p         p          + 1729p         p         p         |
|           {0, 1, 0} {1, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 2, 0} {1, 1,   |
|      -----------------------------------------------------------------------|
|         2               2                   2                               |
|        p          - 721p         p         p          + 82p         p       |
|      0} {1, 1, 1}       {0, 2, 0} {1, 1, 0} {1, 1, 1}      {0, 0, 1} {0, 2, |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p          + 6866p         p         p         p          +|
|      1} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {0, 2, 1} {1, 1, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                                         2                2                  |
|      1083p         p         p         p          + 6868p         p         |
|           {0, 2, 0} {0, 2, 1} {1, 1, 0} {1, 1, 1}        {0, 2, 1} {1, 1,   |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p          + 7228p         p         p         p          - 4392p    |
|      0} {1, 1, 1}        {0, 1, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          + 5688p         p         p        |
|      2, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1}        {0, 2, 1} {1, 0, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|         2                2                   2                              |
|        p          + 1018p         p         p          - 1306p         p    |
|      0} {1, 1, 1}        {1, 0, 0} {1, 1, 0} {1, 1, 1}        {0, 0, 0} {1, |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          + 9747p         p         p         p        |
|      0, 1} {1, 1, 0} {1, 1, 1}        {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|         + 5432p         p         p         p          + 5929p         p    |
|      1}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 2, 0} {1, |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          - 2974p         p         p         p        |
|      0, 1} {1, 1, 0} {1, 1, 1}        {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                               2               2             |
|         + 11893p         p         p         p          - 398p         p    |
|      1}         {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}       {1, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|            2                           2         2                          |
|           p          + 13258p         p         p          - 4952p          |
|      1, 0} {1, 1, 1}         {0, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 1,    |
|      -----------------------------------------------------------------------|
|         2         2                           2         2                   |
|        p         p          - 14053p         p         p          + 2025p   |
|      0} {1, 1, 0} {1, 1, 1}         {0, 2, 0} {1, 1, 0} {1, 1, 1}        {0,|
|      -----------------------------------------------------------------------|
|            2         2                          2         2                 |
|           p         p          + 3102p         p         p          +       |
|      2, 1} {1, 1, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 0} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|                     2         2                3         2                  |
|      2662p         p         p          - 6950p         p          +        |
|           {1, 0, 1} {1, 1, 0} {1, 1, 1}        {1, 1, 0} {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|           2         3                                    3                  |
|      4780p         p          - 1517p         p         p          -        |
|           {0, 0, 1} {1, 1, 1}        {0, 0, 1} {0, 1, 0} {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|           2         3                                    3                  |
|      2844p         p          + 8139p         p         p          +        |
|           {0, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 1, 1} {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|                               3                2         3                  |
|      6285p         p         p          + 6976p         p          +        |
|           {0, 1, 0} {0, 1, 1} {1, 1, 1}        {0, 1, 1} {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|                               3                                    3        |
|      6025p         p         p          - 6581p         p         p         |
|           {0, 0, 1} {0, 2, 0} {1, 1, 1}        {0, 1, 1} {0, 2, 0} {1, 1, 1}|
|      -----------------------------------------------------------------------|
|              2         3                                     3              |
|      - 14064p         p          + 12786p         p         p          -    |
|              {0, 2, 0} {1, 1, 1}         {0, 0, 1} {0, 2, 1} {1, 1, 1}      |
|      -----------------------------------------------------------------------|
|                               3                                    3        |
|      4167p         p         p          - 4571p         p         p         |
|           {0, 1, 0} {0, 2, 1} {1, 1, 1}        {0, 1, 1} {0, 2, 1} {1, 1, 1}|
|      -----------------------------------------------------------------------|
|                                 3               2         3                 |
|      + 3636p         p         p          + 382p         p          +       |
|             {0, 2, 0} {0, 2, 1} {1, 1, 1}       {0, 2, 1} {1, 1, 1}         |
|      -----------------------------------------------------------------------|
|                                3                                     3      |
|      14415p         p         p          - 11560p         p         p       |
|            {0, 1, 0} {1, 0, 0} {1, 1, 1}         {0, 1, 1} {1, 0, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                    3                                    3   |
|         - 8147p         p         p          + 6218p         p         p    |
|      1}        {0, 2, 0} {1, 0, 0} {1, 1, 1}        {0, 2, 1} {1, 0, 0} {1, |
|      -----------------------------------------------------------------------|
|                    2         3                                    3         |
|            - 12311p         p          + 5397p         p         p          |
|      1, 1}         {1, 0, 0} {1, 1, 1}        {0, 0, 0} {1, 0, 1} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|                                 3                                    3      |
|      - 6581p         p         p          - 2784p         p         p       |
|             {0, 0, 1} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|                                     3                                    3  |
|         - 15531p         p         p          - 1756p         p         p   |
|      1}         {0, 1, 1} {1, 0, 1} {1, 1, 1}        {0, 2, 0} {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                                        3                                    |
|            - 10663p         p         p          - 1458p         p          |
|      1, 1}         {0, 2, 1} {1, 0, 1} {1, 1, 1}        {1, 0, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|         3                 2         3                                       |
|        p          - 15294p         p          + 10548p         p            |
|      1} {1, 1, 1}         {1, 0, 1} {1, 1, 1}         {0, 0, 1} {1, 1,      |
|      -----------------------------------------------------------------------|
|         3                                    3                              |
|        p          + 3053p         p         p          + 5136p         p    |
|      0} {1, 1, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1}        {0, 2, 0} {1, |
|      -----------------------------------------------------------------------|
|            3                                    3                           |
|           p          - 2985p         p         p          - 3881p           |
|      1, 0} {1, 1, 1}        {0, 2, 1} {1, 1, 0} {1, 1, 1}        {1, 0,     |
|      -----------------------------------------------------------------------|
|                   3                                    3                2   |
|        p         p          + 5975p         p         p          + 2135p    |
|      0} {1, 1, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 0} {1, 1, 1}        {1, |
|      -----------------------------------------------------------------------|
|            3                           4                          4         |
|           p          + 13496p         p          - 3444p         p          |
|      1, 0} {1, 1, 1}         {0, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 1, 1} |
|      -----------------------------------------------------------------------|
|                       4                          4                          |
|      + 9761p         p          - 5940p         p          - 3851p          |
|             {0, 1, 1} {1, 1, 1}        {0, 2, 0} {1, 1, 1}        {0, 2,    |
|      -----------------------------------------------------------------------|
|         4                          4                          4             |
|        p          - 9475p         p          - 5712p         p          +   |
|      1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}     |
|      -----------------------------------------------------------------------|
|                     4                                                       |
|      1907p         p          - 10265p         p         p         p        |
|           {1, 1, 0} {1, 1, 1}         {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                          2                                                  |
|        p          - 6010p         p         p         p          + 3952p    |
|      0} {1, 2, 0}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}        {0, |
|      -----------------------------------------------------------------------|
|                                                           2                 |
|           p         p         p         p          - 9323p         p        |
|      0, 1} {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}        {0, 2, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 2775p         p         p         p           |
|      1} {1, 1, 0} {1, 2, 0}        {0, 2, 0} {1, 0, 0} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                           2                                                 |
|        p          - 12739p         p         p         p          +         |
|      0} {1, 2, 0}         {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}           |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|      15368p         p         p         p          + 12655p         p       |
|            {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}         {0, 0, 1} {1, 0, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          + 9260p         p         p         p          +|
|      1} {1, 1, 0} {1, 2, 0}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|                     2                                               2       |
|      5782p         p         p         p          + 12079p         p        |
|           {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}         {1, 0, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                    3                                     2  |
|        p         p          + 4213p         p         p          - 12420p   |
|      1} {1, 1, 0} {1, 2, 0}        {1, 0, 1} {1, 1, 0} {1, 2, 0}         {0,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          + 13944p         p         p         p       |
|      0, 1} {1, 1, 0} {1, 2, 0}         {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         - 9109p         p         p         p          - 4584p         p    |
|      0}        {0, 0, 1} {0, 2, 0} {1, 1, 0} {1, 2, 0}        {0, 0, 0} {1, |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          + 10620p         p         p         p       |
|      0, 1} {1, 1, 0} {1, 2, 0}         {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         + 1389p         p         p         p          - 4007p         p    |
|      0}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}        {0, 2, 0} {1, |
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          + 8445p         p         p         p        |
|      0, 1} {1, 1, 0} {1, 2, 0}        {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|               2         2                                     3             |
|         - 231p         p         p          - 10359p         p         p    |
|      0}       {1, 0, 1} {1, 1, 0} {1, 2, 0}         {0, 0, 1} {1, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|                             3                                    2          |
|            + 5689p         p         p          + 8566p         p           |
|      2, 0}        {1, 0, 1} {1, 1, 0} {1, 2, 0}        {0, 0, 1} {0, 1,     |
|      -----------------------------------------------------------------------|
|                                    3                                     2  |
|        p         p          - 4571p         p         p          + 10783p   |
|      0} {1, 1, 1} {1, 2, 0}        {0, 1, 0} {1, 1, 1} {1, 2, 0}         {0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p          + 13326p         p         p       |
|      0, 1} {0, 1, 1} {1, 1, 1} {1, 2, 0}         {0, 0, 1} {0, 1, 0} {0, 1, |
|      -----------------------------------------------------------------------|
|                                   2                                         |
|        p         p          + 507p         p         p         p          - |
|      1} {1, 1, 1} {1, 2, 0}       {0, 1, 0} {0, 1, 1} {1, 1, 1} {1, 2, 0}   |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|      15057p         p         p         p          + 13705p         p       |
|            {0, 0, 1} {0, 1, 1} {1, 1, 1} {1, 2, 0}         {0, 1, 0} {0, 1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 13722p         p         p         p          |
|      1} {1, 1, 1} {1, 2, 0}         {0, 0, 1} {0, 1, 1} {0, 2, 0} {1, 1,    |
|      -----------------------------------------------------------------------|
|                          2                                                  |
|        p          - 7418p         p         p         p          - 2537p    |
|      1} {1, 2, 0}        {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2, 0}        {0, |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p         p          - 11973p         p         p       |
|      0, 1} {0, 2, 0} {1, 1, 1} {1, 2, 0}         {0, 1, 1} {0, 2, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                          2                                              2   |
|        p          + 7766p         p         p         p          + 7907p    |
|      1} {1, 2, 0}        {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 0}        {0, |
|      -----------------------------------------------------------------------|
|                                                            2                |
|           p         p         p          - 10180p         p         p       |
|      2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 0}         {0, 1, 0} {1, 0, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p          + 6712p         p         p         p          + 13696p   |
|      1} {1, 2, 0}        {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 0}         {0,|
|      -----------------------------------------------------------------------|
|            2                                    3                           |
|           p         p         p          - 7989p         p         p        |
|      2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                 2                                                           |
|         + 15438p         p         p         p          + 2428p         p   |
|      0}         {0, 0, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|                                                  2                          |
|           p         p         p          + 11093p         p         p       |
|      1, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}         {0, 1, 0} {1, 0, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 12952p         p         p         p         p          |
|      1} {1, 2, 0}         {0, 0, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      - 2046p         p         p         p         p          + 6204p       |
|             {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, 1, |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p         p          + 3804p         p           |
|      0} {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, 1, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 9004p         p         p         p           |
|      1} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {0, 2, 0} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 13289p         p         p         p         p          |
|      1} {1, 2, 0}         {0, 1, 1} {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|              2                                                              |
|      + 15458p         p         p         p          - 3295p         p      |
|              {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, 1, 0} {1, 0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 35p         p         p         p   |
|      0} {1, 0, 1} {1, 1, 1} {1, 2, 0}      {0, 1, 1} {1, 0, 0} {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p          + 2724p         p         p         p         p        |
|      1, 1} {1, 2, 0}        {0, 2, 0} {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                2                                                         2  |
|         - 2428p         p         p         p          - 11083p         p   |
|      0}        {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}         {0, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p         p          - 6975p         p         p         p        |
|      0, 1} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {1, 0, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                          2                                              2   |
|         - 2858p         p         p         p          + 7911p         p    |
|      0}        {0, 1, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, 1, 1} {1, |
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p         p          - 1055p         p         p         p        |
|      0, 1} {1, 1, 1} {1, 2, 0}        {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                          2                                    3             |
|         - 7263p         p         p         p          + 8231p         p    |
|      0}        {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {1, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|                             2                                               |
|           p          - 7060p         p         p         p          -       |
|      1, 1} {1, 2, 0}        {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}         |
|      -----------------------------------------------------------------------|
|                                                                    2        |
|      3996p         p         p         p         p          - 6935p         |
|           {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 1,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 4845p         p         p           |
|      0} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {0, 2, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          + 3764p         p         p         p          +|
|      0} {1, 1, 1} {1, 2, 0}        {0, 2, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      11028p         p         p         p         p          + 15607p       |
|            {0, 1, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {0, 2, |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p         p          + 8551p         p           |
|      0} {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 4551p         p         p         p           |
|      0} {1, 1, 1} {1, 2, 0}        {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 8932p         p         p         p         p          -|
|      1} {1, 2, 0}        {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      14900p         p         p         p         p          - 14092p       |
|            {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {0, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          - 838p         p            |
|      0} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}       {1, 0, 0} {1, 0,      |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p         p          + 3289p         p         p           |
|      1} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {1, 0, 1} {1, 1, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          + 11964p         p         p         p          + 644p    |
|      1} {1, 2, 0}         {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}       {0, |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p         p          - 13395p         p         p       |
|      1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {0, 2, 0} {1, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p          + 9272p         p         p         p          - 3486p    |
|      1} {1, 2, 0}        {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {1, |
|      -----------------------------------------------------------------------|
|            2                                     3                          |
|           p         p         p          - 15866p         p         p       |
|      0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {1, 1, 0} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                2         2                                             2    |
|         + 4083p         p         p          + 590p         p         p     |
|      0}        {0, 0, 1} {1, 1, 1} {1, 2, 0}       {0, 0, 1} {0, 1, 0} {1,  |
|      -----------------------------------------------------------------------|
|                             2         2                                     |
|           p          + 1965p         p         p          + 2029p           |
|      1, 1} {1, 2, 0}        {0, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 0,     |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p         p          - 7645p         p         p           |
|      1} {0, 1, 1} {1, 1, 1} {1, 2, 0}        {0, 1, 0} {0, 1, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                          2         2                                        |
|        p          + 4635p         p         p          + 10092p         p   |
|      1} {1, 2, 0}        {0, 1, 1} {1, 1, 1} {1, 2, 0}         {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          + 7761p         p         p         p        |
|      2, 0} {1, 1, 1} {1, 2, 0}        {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                 2         2                                                 |
|         + 14688p         p         p          + 14932p         p            |
|      0}         {0, 2, 0} {1, 1, 1} {1, 2, 0}         {0, 1, 0} {1, 0,      |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p         p          - 10522p         p         p         p          |
|      0} {1, 1, 1} {1, 2, 0}         {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|                                  2                          2         2     |
|      + 12743p         p         p         p          - 2014p         p      |
|              {0, 2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 1,|
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          + 1311p         p         p         p          - 7865p    |
|      1} {1, 2, 0}        {0, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p         p          - 5377p         p         p        |
|      0, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, 1, 0} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p          - 10326p         p         p         p          -         |
|      1} {1, 2, 0}         {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0}           |
|      -----------------------------------------------------------------------|
|                                2                                            |
|      15838p         p         p         p          + 2393p         p        |
|            {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|         2                           2         2                             |
|        p         p          - 12849p         p         p          - 7153p   |
|      1} {1, 1, 1} {1, 2, 0}         {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p         p          - 9284p         p         p        |
|      0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 1, 0} {1, 1, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p          + 13559p         p         p         p          + 6295p   |
|      1} {1, 2, 0}         {0, 2, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {1,|
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p         p          - 10828p         p         p       |
|      0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}         {1, 0, 1} {1, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                           2         2                                    3  |
|        p          + 13001p         p         p          - 8825p         p   |
|      1} {1, 2, 0}         {1, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                                        3                                    |
|           p          - 14990p         p         p          - 2506p          |
|      1, 1} {1, 2, 0}         {0, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 1,    |
|      -----------------------------------------------------------------------|
|         3                                    3                              |
|        p         p          + 7346p         p         p          - 11764p   |
|      1} {1, 1, 1} {1, 2, 0}        {0, 2, 0} {1, 1, 1} {1, 2, 0}         {1,|
|      -----------------------------------------------------------------------|
|            3                                    3                           |
|           p         p          + 3590p         p         p          +       |
|      0, 0} {1, 1, 1} {1, 2, 0}        {1, 0, 1} {1, 1, 1} {1, 2, 0}         |
|      -----------------------------------------------------------------------|
|                    3                          4                          2  |
|      176p         p         p          - 1513p         p          + 8455p   |
|          {1, 1, 0} {1, 1, 1} {1, 2, 0}        {1, 1, 1} {1, 2, 0}        {0,|
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p          + 11936p         p         p         p       |
|      0, 1} {1, 1, 0} {1, 2, 0}         {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|         + 2396p         p         p         p          - 13654p         p   |
|      0}        {0, 0, 1} {0, 2, 0} {1, 1, 0} {1, 2, 0}         {0, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p          + 14484p         p         p         p       |
|      0, 1} {1, 1, 0} {1, 2, 0}         {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                               2                             |
|         + 11669p         p         p         p          + 15128p            |
|      0}         {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0}         {0, 2,      |
|      -----------------------------------------------------------------------|
|                             2                                               |
|        p         p         p          - 11005p         p         p          |
|      0} {1, 0, 1} {1, 1, 0} {1, 2, 0}         {1, 0, 0} {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|         2                2                   2                        2     |
|        p          - 4868p         p         p          - 93p         p      |
|      0} {1, 2, 0}        {1, 0, 1} {1, 1, 0} {1, 2, 0}      {0, 0, 1} {1, 1,|
|      -----------------------------------------------------------------------|
|         2                          2         2                2             |
|        p          - 5146p         p         p          - 3154p         p    |
|      0} {1, 2, 0}        {1, 0, 1} {1, 1, 0} {1, 2, 0}        {0, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p          + 2198p         p         p         p          +       |
|      1, 1} {1, 2, 0}        {0, 0, 1} {0, 1, 0} {1, 1, 1} {1, 2, 0}         |
|      -----------------------------------------------------------------------|
|            2                   2                                            |
|      13521p         p         p          - 6543p         p         p        |
|            {0, 1, 0} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {0, 1, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|         2                                               2                2  |
|        p          - 13593p         p         p         p          - 3993p   |
|      1} {1, 2, 0}         {0, 1, 0} {0, 1, 1} {1, 1, 1} {1, 2, 0}        {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          + 6513p         p         p         p        |
|      1, 1} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {0, 2, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                               2                 2           |
|         - 11797p         p         p         p          - 15269p            |
|      0}         {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2, 0}         {0, 2,      |
|      -----------------------------------------------------------------------|
|                   2                                               2         |
|        p         p          + 13971p         p         p         p          |
|      0} {1, 1, 1} {1, 2, 0}         {0, 1, 0} {1, 0, 0} {1, 1, 1} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|                                            2                                |
|      - 12296p         p         p         p          - 14406p         p     |
|              {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 0}         {0, 2, 0} {1,  |
|      -----------------------------------------------------------------------|
|                      2                2                   2                 |
|           p         p          + 1930p         p         p          -       |
|      0, 0} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 1, 1} {1, 2, 0}         |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      12109p         p         p         p          + 11111p         p       |
|            {0, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}         {0, 0, 1} {1, 0, |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p          + 9038p         p         p         p          -|
|      1} {1, 1, 1} {1, 2, 0}        {0, 1, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0}  |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      12876p         p         p         p          + 11978p         p       |
|            {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 0}         {0, 2, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|                   2                                               2         |
|        p         p          + 11168p         p         p         p          |
|      1} {1, 1, 1} {1, 2, 0}         {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|             2                   2                                           |
|      + 1000p         p         p          + 4434p         p         p       |
|             {1, 0, 1} {1, 1, 1} {1, 2, 0}        {0, 0, 1} {1, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p          + 15632p         p         p         p          + 2767p   |
|      1} {1, 2, 0}         {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {0,|
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          - 4497p         p         p        |
|      2, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 1, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|         2                                              2                2   |
|        p          - 1602p         p         p         p          + 1232p    |
|      1} {1, 2, 0}        {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 0}        {1, |
|      -----------------------------------------------------------------------|
|                      2                         2         2                  |
|           p         p          - 464p         p         p          + 338p   |
|      1, 0} {1, 1, 1} {1, 2, 0}       {0, 0, 1} {1, 1, 1} {1, 2, 0}       {0,|
|      -----------------------------------------------------------------------|
|            2         2                           2         2                |
|           p         p          - 11183p         p         p          -      |
|      1, 0} {1, 1, 1} {1, 2, 0}         {0, 1, 1} {1, 1, 1} {1, 2, 0}        |
|      -----------------------------------------------------------------------|
|                     2         2                           2         2       |
|      6046p         p         p          - 14565p         p         p        |
|           {0, 2, 0} {1, 1, 1} {1, 2, 0}         {1, 0, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                          2         2                         2         2    |
|         - 5417p         p         p          + 539p         p         p     |
|      0}        {1, 0, 1} {1, 1, 1} {1, 2, 0}       {1, 1, 0} {1, 1, 1} {1,  |
|      -----------------------------------------------------------------------|
|                   3         2                                     3         |
|            + 6966p         p          + 10201p         p         p          |
|      2, 0}        {1, 1, 1} {1, 2, 0}         {0, 0, 1} {1, 1, 0} {1, 2, 0} |
|      -----------------------------------------------------------------------|
|                                  3                                    3     |
|      - 14397p         p         p          + 6104p         p         p      |
|              {1, 0, 1} {1, 1, 0} {1, 2, 0}        {0, 0, 1} {1, 1, 1} {1, 2,|
|      -----------------------------------------------------------------------|
|                                    3                                     3  |
|         - 4966p         p         p          + 13594p         p         p   |
|      0}        {0, 1, 0} {1, 1, 1} {1, 2, 0}         {0, 1, 1} {1, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|                                       3                                     |
|            - 2135p         p         p          + 5551p         p           |
|      2, 0}        {0, 2, 0} {1, 1, 1} {1, 2, 0}        {1, 0, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|         3                                    3                              |
|        p          - 3848p         p         p          + 14470p         p   |
|      1} {1, 2, 0}        {1, 0, 1} {1, 1, 1} {1, 2, 0}         {1, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|            3               2         3                          2           |
|           p          - 446p         p          - 1353p         p            |
|      1, 1} {1, 2, 0}       {1, 1, 1} {1, 2, 0}        {0, 0, 1} {0, 1,      |
|      -----------------------------------------------------------------------|
|                                    3                                        |
|        p         p          - 9174p         p         p          - 2221p    |
|      1} {0, 2, 0} {1, 2, 1}        {0, 1, 1} {0, 2, 0} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                      2                          2         2                 |
|           p         p         p          + 4000p         p         p        |
|      0, 1} {0, 1, 1} {0, 2, 0} {1, 2, 1}        {0, 1, 1} {0, 2, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                          3                                     3            |
|         - 9551p         p         p          - 10903p         p         p   |
|      1}        {0, 0, 1} {0, 2, 0} {1, 2, 1}         {0, 1, 1} {0, 2, 0} {1,|
|      -----------------------------------------------------------------------|
|                    2                                                        |
|            - 14338p         p         p         p          + 5391p          |
|      2, 1}         {0, 0, 1} {0, 1, 1} {0, 2, 1} {1, 2, 1}        {0, 0,    |
|      -----------------------------------------------------------------------|
|                                                                   2         |
|        p         p         p         p          - 10955p         p          |
|      1} {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 2, 1}         {0, 0, 1} {0, 1,    |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          - 11109p         p         p         p          |
|      1} {0, 2, 1} {1, 2, 1}         {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|             3                                                               |
|      - 8039p         p         p          + 11510p         p         p      |
|             {0, 1, 1} {0, 2, 1} {1, 2, 1}         {0, 0, 1} {0, 1, 1} {0, 2,|
|      -----------------------------------------------------------------------|
|                                   2                                         |
|        p         p          + 543p         p         p         p          - |
|      0} {0, 2, 1} {1, 2, 1}       {0, 1, 1} {0, 2, 0} {0, 2, 1} {1, 2, 1}   |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|      12061p         p         p         p          - 5093p         p        |
|            {0, 0, 1} {0, 2, 0} {0, 2, 1} {1, 2, 1}        {0, 1, 1} {0, 2,  |
|      -----------------------------------------------------------------------|
|                                    3                                    2   |
|        p         p          - 2404p         p         p          - 4323p    |
|      0} {0, 2, 1} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          + 5257p         p         p         p        |
|      0, 1} {0, 2, 1} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {0, 2, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         - 7119p         p         p         p          - 15560p         p   |
|      1}        {0, 0, 1} {0, 1, 1} {0, 2, 1} {1, 2, 1}         {0, 1, 0} {0,|
|      -----------------------------------------------------------------------|
|            2                           2         2                          |
|           p         p          + 12761p         p         p          +      |
|      1, 1} {0, 2, 1} {1, 2, 1}         {0, 1, 1} {0, 2, 1} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                               2                                             |
|      1334p         p         p         p          - 3667p         p         |
|           {0, 0, 1} {0, 2, 0} {0, 2, 1} {1, 2, 1}        {0, 1, 1} {0, 2,   |
|      -----------------------------------------------------------------------|
|         2                          2         2                              |
|        p         p          - 3971p         p         p          - 15230p   |
|      0} {0, 2, 1} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            3                                     3                          |
|           p         p          + 12633p         p         p          +      |
|      0, 1} {0, 2, 1} {1, 2, 1}         {0, 1, 0} {0, 2, 1} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                     3                                     3                 |
|      7939p         p         p          + 13184p         p         p        |
|           {0, 1, 1} {0, 2, 1} {1, 2, 1}         {0, 2, 0} {0, 2, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                2                                              2             |
|         + 2115p         p         p         p          - 8474p         p    |
|      1}        {0, 1, 1} {0, 2, 1} {1, 0, 0} {1, 2, 1}        {0, 2, 0} {0, |
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p         p          + 8066p         p         p         p        |
|      2, 1} {1, 0, 0} {1, 2, 1}        {0, 1, 0} {0, 2, 1} {1, 0, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                          2                                               2  |
|         + 9324p         p         p         p          - 12297p         p   |
|      1}        {0, 1, 1} {0, 2, 1} {1, 0, 0} {1, 2, 1}         {0, 2, 0} {0,|
|      -----------------------------------------------------------------------|
|                                        3                                    |
|           p         p          + 15653p         p         p          +      |
|      2, 1} {1, 0, 0} {1, 2, 1}         {0, 2, 1} {1, 0, 0} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|           2         2                                              2        |
|      2550p         p         p          + 4862p         p         p         |
|           {0, 2, 0} {1, 0, 0} {1, 2, 1}        {0, 1, 0} {0, 2, 1} {1, 0,   |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          - 5575p         p         p         p          + 14230p   |
|      0} {1, 2, 1}        {0, 1, 1} {0, 2, 1} {1, 0, 0} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|                      2                           2         2                |
|           p         p         p          - 14217p         p         p       |
|      2, 0} {0, 2, 1} {1, 0, 0} {1, 2, 1}         {0, 2, 1} {1, 0, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                          3                                     3            |
|         + 8502p         p         p          - 11822p         p         p   |
|      1}        {0, 2, 0} {1, 0, 0} {1, 2, 1}         {0, 2, 1} {1, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|               2                                                             |
|            + p         p         p         p          + 613p         p      |
|      2, 1}    {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 2, 1}       {0, 0, 1} {0, 1,|
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p          - 1591p         p         p           |
|      0} {0, 1, 1} {1, 0, 1} {1, 2, 1}        {0, 0, 0} {0, 1, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p          - 6768p         p         p         p          + 14622p   |
|      1} {1, 2, 1}        {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2                                    2                           |
|           p         p         p          + 7169p         p         p        |
|      1, 0} {0, 1, 1} {1, 0, 1} {1, 2, 1}        {0, 1, 1} {0, 2, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p          - 3647p         p         p         p          + 11770p   |
|      1} {1, 2, 1}        {0, 0, 1} {0, 2, 0} {1, 0, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2                                     3                          |
|           p         p         p          + 12422p         p         p       |
|      1, 1} {0, 2, 0} {1, 0, 1} {1, 2, 1}         {0, 2, 0} {1, 0, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                2                                                            |
|         - 7579p         p         p         p          - 1561p         p    |
|      1}        {0, 0, 1} {0, 2, 1} {1, 0, 1} {1, 2, 1}        {0, 0, 1} {0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p          - 5097p         p         p        |
|      1, 0} {0, 2, 1} {1, 0, 1} {1, 2, 1}        {0, 0, 0} {0, 1, 1} {0, 2,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 4364p         p         p         p           |
|      1} {1, 0, 1} {1, 2, 1}        {0, 0, 1} {0, 1, 1} {0, 2, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 14867p         p         p         p         p          |
|      1} {1, 2, 1}         {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 0, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|             2                                                               |
|      + 6741p         p         p         p          - 1281p         p       |
|             {0, 1, 1} {0, 2, 1} {1, 0, 1} {1, 2, 1}        {0, 0, 1} {0, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 11733p         p         p          |
|      0} {0, 2, 1} {1, 0, 1} {1, 2, 1}         {0, 1, 1} {0, 2, 0} {0, 2,    |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p         p          - 13582p         p         p         p          |
|      1} {1, 0, 1} {1, 2, 1}         {0, 2, 0} {0, 2, 1} {1, 0, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                       2                                              2      |
|      + 8462p         p         p         p          - 7119p         p       |
|             {0, 0, 0} {0, 2, 1} {1, 0, 1} {1, 2, 1}        {0, 0, 1} {0, 2, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 7426p         p         p         p          +|
|      1} {1, 0, 1} {1, 2, 1}        {0, 1, 0} {0, 2, 1} {1, 0, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                              2        |
|      1091p         p         p         p          + 9519p         p         |
|           {0, 1, 1} {0, 2, 1} {1, 0, 1} {1, 2, 1}        {0, 2, 0} {0, 2,   |
|      -----------------------------------------------------------------------|
|                                   3                                    2    |
|        p         p          + 924p         p         p          + 3022p     |
|      1} {1, 0, 1} {1, 2, 1}       {0, 2, 1} {1, 0, 1} {1, 2, 1}        {0,  |
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p         p         p          + 6954p         p         p        |
|      1, 1} {1, 0, 0} {1, 0, 1} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 4275p         p         p         p         p          -|
|      1} {1, 2, 1}        {0, 1, 0} {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      11652p         p         p         p         p          + 14515p       |
|            {0, 1, 1} {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 2, 1}         {0, 2, |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p         p          - 1255p         p           |
|      0} {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 2, 1}        {0, 2, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 1141p         p         p         p          -|
|      0} {1, 0, 1} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 0, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                                       |
|      1745p         p         p         p          - 6142p         p         |
|           {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 2, 1}        {0, 0, 0} {0, 1,   |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p         p          - 12533p         p         p         p          |
|      1} {1, 0, 1} {1, 2, 1}         {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                  2                           2         2    |
|      + 14086p         p         p         p          + 13723p         p     |
|              {0, 1, 0} {0, 1, 1} {1, 0, 1} {1, 2, 1}         {0, 1, 1} {1,  |
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p          - 4551p         p         p         p          -       |
|      0, 1} {1, 2, 1}        {0, 0, 1} {0, 2, 0} {1, 0, 1} {1, 2, 1}         |
|      -----------------------------------------------------------------------|
|                                2                          2         2       |
|      11980p         p         p         p          + 9766p         p        |
|            {0, 1, 1} {0, 2, 0} {1, 0, 1} {1, 2, 1}        {0, 2, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          - 4855p         p         p         p          - 2933p    |
|      1} {1, 2, 1}        {0, 0, 0} {0, 2, 1} {1, 0, 1} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p         p          - 10793p         p         p       |
|      0, 1} {0, 2, 1} {1, 0, 1} {1, 2, 1}         {0, 1, 0} {0, 2, 1} {1, 0, |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p          - 12172p         p         p         p          - 801p    |
|      1} {1, 2, 1}         {0, 1, 1} {0, 2, 1} {1, 0, 1} {1, 2, 1}       {0, |
|      -----------------------------------------------------------------------|
|                      2                           2         2                |
|           p         p         p          + 13118p         p         p       |
|      2, 0} {0, 2, 1} {1, 0, 1} {1, 2, 1}         {0, 2, 1} {1, 0, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|         - 11620p         p         p         p          - 6291p         p   |
|      1}         {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 2, 1}        {0, 2, 0} {1,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          + 4422p         p         p         p        |
|      0, 0} {1, 0, 1} {1, 2, 1}        {0, 2, 1} {1, 0, 0} {1, 0, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                           3                                     3           |
|         + 11552p         p         p          - 15406p         p            |
|      1}         {0, 2, 0} {1, 0, 1} {1, 2, 1}         {0, 2, 1} {1, 0,      |
|      -----------------------------------------------------------------------|
|                                    2                                     3  |
|        p          + 4387p         p         p         p          + 11760p   |
|      1} {1, 2, 1}        {0, 0, 1} {0, 2, 0} {1, 1, 0} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|                                        2                                    |
|           p         p          + 15505p         p         p         p       |
|      2, 0} {1, 1, 0} {1, 2, 1}         {0, 0, 1} {0, 2, 1} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                                                          2  |
|         - 6671p         p         p         p         p          - 10166p   |
|      1}        {0, 0, 1} {0, 1, 0} {0, 2, 1} {1, 1, 0} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p          + 14656p         p         p       |
|      1, 0} {0, 2, 1} {1, 1, 0} {1, 2, 1}         {0, 0, 1} {0, 2, 0} {0, 2, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          + 4012p         p         p         p          -|
|      1} {1, 1, 0} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 1, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|      12731p         p         p         p          + 3920p         p        |
|            {0, 0, 1} {0, 2, 1} {1, 1, 0} {1, 2, 1}        {0, 1, 0} {0, 2,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          + 1208p         p         p         p          +|
|      1} {1, 1, 0} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 1, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|           3                                                                 |
|      3959p         p         p          - 5316p         p         p         |
|           {0, 2, 1} {1, 1, 0} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 0,   |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p         p          + 12054p         p         p         p          |
|      0} {1, 1, 0} {1, 2, 1}         {0, 2, 1} {1, 0, 0} {1, 1, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                        2                                               2    |
|      + 15838p         p         p         p          + 12717p         p     |
|              {0, 1, 0} {1, 0, 0} {1, 1, 0} {1, 2, 1}         {0, 2, 0} {1,  |
|      -----------------------------------------------------------------------|
|                                                  2                          |
|           p         p          + 14158p         p         p         p       |
|      0, 0} {1, 1, 0} {1, 2, 1}         {0, 2, 1} {1, 0, 0} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                 3                                    2                      |
|         + 12453p         p         p          - 7456p         p         p   |
|      1}         {1, 0, 0} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p          - 5955p         p         p         p         p        |
|      1, 0} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                 2                                                           |
|         + 15284p         p         p         p          - 8352p         p   |
|      1}         {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p         p         p          + 8420p         p         p        |
|      2, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 2, 0} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 13696p         p         p         p         p          |
|      0} {1, 2, 1}         {0, 0, 0} {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      - 2923p         p         p         p         p          - 5797p       |
|             {0, 0, 1} {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          + 7029p         p           |
|      0} {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 2, 0} {0, 2,     |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p         p          - 10111p         p         p          |
|      1} {1, 0, 1} {1, 1, 0} {1, 2, 1}         {0, 2, 1} {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 562p         p         p         p         p          + |
|      0} {1, 2, 1}       {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}   |
|      -----------------------------------------------------------------------|
|                                                                             |
|      2744p         p         p         p         p          - 7906p         |
|           {0, 2, 0} {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 2,   |
|      -----------------------------------------------------------------------|
|                                                         2                   |
|        p         p         p         p          - 10891p         p          |
|      1} {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}         {1, 0, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          + 10254p         p         p         p          |
|      1} {1, 1, 0} {1, 2, 1}         {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                     2                                               2       |
|      + 43p         p         p         p          - 13396p         p        |
|           {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 2, 1}         {0, 1, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          - 13566p         p         p         p          |
|      1} {1, 1, 0} {1, 2, 1}         {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                        2                                              2     |
|      - 15179p         p         p         p          + 1001p         p      |
|              {0, 2, 1} {1, 0, 1} {1, 1, 0} {1, 2, 1}        {1, 0, 0} {1, 0,|
|      -----------------------------------------------------------------------|
|                                   3                                    2    |
|        p         p          - 404p         p         p          + 2647p     |
|      1} {1, 1, 0} {1, 2, 1}       {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0,  |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          - 12259p         p         p         p       |
|      0, 1} {1, 1, 0} {1, 2, 1}         {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                 2         2                                              2  |
|         + 15173p         p         p          - 6164p         p         p   |
|      1}         {0, 1, 0} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {0, 2, 0} {1,|
|      -----------------------------------------------------------------------|
|                             2         2                                     |
|           p          - 4860p         p         p          + 10317p          |
|      1, 0} {1, 2, 1}        {0, 2, 0} {1, 1, 0} {1, 2, 1}         {0, 0,    |
|      -----------------------------------------------------------------------|
|                   2                                               2         |
|        p         p         p          + 15814p         p         p          |
|      1} {0, 2, 1} {1, 1, 0} {1, 2, 1}         {0, 1, 0} {0, 2, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                              2                           2  |
|        p          - 7492p         p         p         p          + 10349p   |
|      0} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 1, 0} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          - 2950p         p         p         p        |
|      2, 1} {1, 1, 0} {1, 2, 1}        {0, 1, 0} {1, 0, 0} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         + 1132p         p         p         p          + 289p         p     |
|      1}        {0, 2, 0} {1, 0, 0} {1, 1, 0} {1, 2, 1}       {0, 2, 1} {1,  |
|      -----------------------------------------------------------------------|
|            2                          2         2                           |
|           p         p          - 4312p         p         p          +       |
|      0, 0} {1, 1, 0} {1, 2, 1}        {1, 0, 0} {1, 1, 0} {1, 2, 1}         |
|      -----------------------------------------------------------------------|
|                               2                                             |
|      6243p         p         p         p          - 636p         p          |
|           {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}       {0, 0, 1} {1, 0,    |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p         p          + 6381p         p         p         p          -|
|      1} {1, 1, 0} {1, 2, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                               2                                             |
|      6065p         p         p         p          - 14098p         p        |
|           {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}         {0, 2, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p         p          - 2379p         p         p         p          +|
|      1} {1, 1, 0} {1, 2, 1}        {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|            2         2                                    3                 |
|      10287p         p         p          - 3128p         p         p        |
|            {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                          3                                    3             |
|         - 1229p         p         p          + 1826p         p         p    |
|      1}        {0, 1, 0} {1, 1, 0} {1, 2, 1}        {0, 2, 0} {1, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|                             3                                    3          |
|            - 9944p         p         p          + 6040p         p           |
|      2, 1}        {0, 2, 1} {1, 1, 0} {1, 2, 1}        {1, 0, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                    3                          4             |
|        p          + 3527p         p         p          + 7286p         p    |
|      0} {1, 2, 1}        {1, 0, 1} {1, 1, 0} {1, 2, 1}        {1, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|                              2                                    3         |
|            - 15449p         p         p         p          - 3879p          |
|      2, 1}         {0, 0, 1} {0, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 1,    |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          + 3361p         p         p         p          +|
|      0} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 1, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                    2        |
|      11330p         p         p         p         p          + 965p         |
|            {0, 0, 1} {0, 1, 0} {0, 1, 1} {1, 1, 1} {1, 2, 1}       {0, 1,   |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p          + 4196p         p         p           |
|      0} {0, 1, 1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 1, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p          + 6825p         p         p         p          + 6339p    |
|      1} {1, 2, 1}        {0, 1, 0} {0, 1, 1} {1, 1, 1} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                                           2                 |
|           p         p         p         p          + 6969p         p        |
|      0, 1} {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {0, 2,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          + 6197p         p         p         p          +|
|      0} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 2, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                      2                                     3                |
|      11132p         p         p         p          - 11420p         p       |
|            {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2, 1}         {0, 2, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                           2                                                 |
|        p          - 13915p         p         p         p          - 2091p   |
|      1} {1, 2, 1}         {0, 0, 1} {0, 2, 1} {1, 1, 1} {1, 2, 1}        {0,|
|      -----------------------------------------------------------------------|
|                                                           2                 |
|           p         p         p         p          - 7784p         p        |
|      0, 1} {0, 1, 0} {0, 2, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 0} {0, 2,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 12140p         p         p         p          |
|      1} {1, 1, 1} {1, 2, 1}         {0, 0, 1} {0, 1, 1} {0, 2, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 12179p         p         p         p         p          |
|      1} {1, 2, 1}         {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 1, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|              2                                                              |
|      - 15948p         p         p         p          - 11704p         p     |
|              {0, 1, 1} {0, 2, 1} {1, 1, 1} {1, 2, 1}         {0, 0, 1} {0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p          + 5027p         p         p        |
|      2, 0} {0, 2, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {0, 2, 0} {0, 2,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          + 2118p         p         p         p          +|
|      1} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                              2        |
|      9247p         p         p         p          + 2679p         p         |
|           {0, 0, 1} {0, 2, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 0} {0, 2,   |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          + 4229p         p         p         p          +|
|      1} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {0, 2, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                    3                  |
|      2103p         p         p         p          - 5221p         p         |
|           {0, 2, 0} {0, 2, 1} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 1,   |
|      -----------------------------------------------------------------------|
|                          2                                              2   |
|        p          + 5599p         p         p         p          - 2778p    |
|      1} {1, 2, 1}        {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p          - 1568p         p         p        |
|      2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 1}        {0, 1, 0} {0, 2, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 15509p         p         p         p          |
|      0} {1, 1, 1} {1, 2, 1}         {0, 1, 1} {0, 2, 1} {1, 0, 0} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 442p         p         p         p         p          - |
|      1} {1, 2, 1}       {0, 2, 0} {0, 2, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1}   |
|      -----------------------------------------------------------------------|
|            2                                                         2      |
|      15358p         p         p         p          + 15870p         p       |
|            {0, 2, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          + 12194p         p         p         p          |
|      0} {1, 1, 1} {1, 2, 1}         {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                       2                                              2      |
|      - 5147p         p         p         p          - 4501p         p       |
|             {0, 2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 0, |
|      -----------------------------------------------------------------------|
|                                     3                                    2  |
|        p         p          - 11854p         p         p          - 9064p   |
|      0} {1, 1, 1} {1, 2, 1}         {1, 0, 0} {1, 1, 1} {1, 2, 1}        {0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p          + 3236p         p         p        |
|      0, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p         p          - 13016p         p         p         p          |
|      1} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      + 1485p         p         p         p         p          - 513p        |
|             {0, 0, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}       {0, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          - 13168p         p          |
|      1} {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {0, 1,    |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p         p          - 11172p         p         p          |
|      1} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0, 1, 1} {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 5563p         p         p         p         p          -|
|      1} {1, 2, 1}        {0, 0, 1} {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                      2      |
|      13130p         p         p         p         p          + 13630p       |
|            {0, 1, 1} {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 13792p         p         p          |
|      0} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0, 0, 0} {0, 2, 1} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 2179p         p         p         p           |
|      1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 2, 1} {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 7947p         p         p         p         p          +|
|      1} {1, 2, 1}        {0, 1, 0} {0, 2, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      10513p         p         p         p         p          + 4020p        |
|            {0, 1, 1} {0, 2, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 2,  |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p         p          + 8788p         p           |
|      0} {0, 2, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 14420p         p         p         p          |
|      1} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 7810p         p         p         p         p          -|
|      1} {1, 2, 1}        {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      12802p         p         p         p         p          + 15376p       |
|            {0, 2, 0} {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0, 2, |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p         p          + 8666p         p           |
|      1} {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {1, 0, 0} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 5810p         p         p         p          +|
|      1} {1, 1, 1} {1, 2, 1}        {0, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|      15765p         p         p         p          - 13638p         p       |
|            {0, 0, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          - 3336p         p         p         p          +|
|      1} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                              2        |
|      4303p         p         p         p          + 5612p         p         |
|           {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 0,   |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          - 13282p         p         p         p          |
|      1} {1, 1, 1} {1, 2, 1}         {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|              3                                    2                         |
|      + 14207p         p         p          + 1090p         p         p      |
|              {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {1, 1, 0} {1, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 8261p         p         p         p         p          -|
|      1} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|           2                                                                 |
|      6469p         p         p         p          - 2377p         p         |
|           {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 2,   |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p         p          + 13182p         p         p          |
|      0} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 2, 0} {1, 1, 0} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 3835p         p         p         p         p          -|
|      1} {1, 2, 1}        {0, 0, 1} {0, 2, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      11863p         p         p         p         p          - 15900p       |
|            {0, 1, 0} {0, 2, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 2, |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p         p          - 1897p         p           |
|      0} {0, 2, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 6205p         p         p         p           |
|      0} {1, 1, 1} {1, 2, 1}        {0, 1, 0} {1, 0, 0} {1, 1, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 5558p         p         p         p         p          -|
|      1} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                     2       |
|      1701p         p         p         p         p          + 10133p        |
|           {0, 2, 1} {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 209p         p         p            |
|      0} {1, 1, 0} {1, 1, 1} {1, 2, 1}       {0, 0, 0} {1, 0, 1} {1, 1,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 9101p         p         p         p           |
|      0} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 8458p         p         p         p         p          -|
|      1} {1, 2, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      14618p         p         p         p         p          + 11482p       |
|            {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          - 12443p         p          |
|      1} {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {1, 0, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p         p          - 8854p         p         p           |
|      1} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {1, 0, 1} {1, 1, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          + 14517p         p         p         p          +         |
|      1} {1, 2, 1}         {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}           |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|      11149p         p         p         p          - 3288p         p        |
|            {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          + 3447p         p         p         p          -|
|      0} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                      2                                            2         |
|      15078p         p         p         p          + 43p         p          |
|            {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}      {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                    3                                     2  |
|        p         p          + 7367p         p         p          - 12572p   |
|      0} {1, 1, 1} {1, 2, 1}        {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          + 3150p         p         p         p        |
|      0, 1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                 2         2                                                 |
|         - 14415p         p         p          + 10491p         p            |
|      1}         {0, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 0, 1} {0, 1,      |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p         p          - 3548p         p         p         p          -|
|      1} {1, 1, 1} {1, 2, 1}        {0, 1, 0} {0, 1, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|           2         2                                               2       |
|      6486p         p         p          + 13178p         p         p        |
|           {0, 1, 1} {1, 1, 1} {1, 2, 1}         {0, 0, 1} {0, 2, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                              2                           2  |
|        p          + 4337p         p         p         p          + 14836p   |
|      1} {1, 2, 1}        {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          + 13639p         p         p         p       |
|      2, 0} {1, 1, 1} {1, 2, 1}         {0, 0, 1} {0, 2, 1} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|         + 10993p         p         p         p          + 5956p         p   |
|      1}         {0, 1, 0} {0, 2, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {0,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          + 15240p         p         p         p       |
|      2, 1} {1, 1, 1} {1, 2, 1}         {0, 2, 0} {0, 2, 1} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                 2         2                                                 |
|         - 15886p         p         p          - 14296p         p            |
|      1}         {0, 2, 1} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, 0,      |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p         p          + 12182p         p         p         p          |
|      0} {1, 1, 1} {1, 2, 1}         {0, 1, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                 2                                           |
|      - 6884p         p         p         p          + 8827p         p       |
|             {0, 2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 0, |
|      -----------------------------------------------------------------------|
|         2                          2         2                              |
|        p         p          - 6220p         p         p          + 15275p   |
|      0} {1, 1, 1} {1, 2, 1}        {1, 0, 0} {1, 1, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p         p          + 6776p         p         p        |
|      0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          - 5476p         p         p         p          + 5282p    |
|      1} {1, 2, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p         p          - 7765p         p         p        |
|      1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p          - 14214p         p         p         p          - 2527p   |
|      1} {1, 2, 1}         {0, 2, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {1,|
|      -----------------------------------------------------------------------|
|                      2                           2         2                |
|           p         p         p          + 13344p         p         p       |
|      0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {1, 0, 1} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                                   2                                         |
|         + 491p         p         p         p          + 12565p         p    |
|      1}       {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          + 7399p         p         p         p        |
|      1, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {1, 1, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         + 5961p         p         p         p          + 5370p         p    |
|      1}        {0, 2, 1} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {1, 0, 0} {1, |
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          - 7996p         p         p         p        |
|      1, 0} {1, 1, 1} {1, 2, 1}        {1, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                2         2                                     3            |
|         - 1703p         p         p          - 13856p         p         p   |
|      1}        {1, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 0, 1} {1, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|                             3                                     3         |
|            + 4836p         p         p          - 15386p         p          |
|      2, 1}        {0, 1, 0} {1, 1, 1} {1, 2, 1}         {0, 1, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                    3                                    3   |
|        p          - 1152p         p         p          + 6478p         p    |
|      1} {1, 2, 1}        {0, 2, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, |
|      -----------------------------------------------------------------------|
|                                        3                                    |
|           p          - 14800p         p         p          - 12442p         |
|      1, 1} {1, 2, 1}         {1, 0, 0} {1, 1, 1} {1, 2, 1}         {1, 0,   |
|      -----------------------------------------------------------------------|
|         3                                     3                             |
|        p         p          + 10244p         p         p          -         |
|      1} {1, 1, 1} {1, 2, 1}         {1, 1, 0} {1, 1, 1} {1, 2, 1}           |
|      -----------------------------------------------------------------------|
|            4                           2                                    |
|      11661p         p          - 15402p         p         p         p       |
|            {1, 1, 1} {1, 2, 1}         {0, 0, 1} {0, 1, 1} {1, 2, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                                                             |
|         - 6035p         p         p         p         p          + 9468p    |
|      1}        {0, 0, 1} {0, 1, 0} {0, 1, 1} {1, 2, 0} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p         p          + 14192p         p         p       |
|      0, 1} {0, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 1, 0} {0, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                           3                                                 |
|        p          + 11469p         p         p          + 5527p         p   |
|      0} {1, 2, 1}         {0, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p         p         p          - 4661p         p         p        |
|      1, 1} {0, 2, 0} {1, 2, 0} {1, 2, 1}        {0, 1, 1} {0, 2, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p          - 3368p         p         p         p          - 11659p   |
|      0} {1, 2, 1}        {0, 0, 1} {0, 2, 0} {1, 2, 0} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2                                    2                           |
|           p         p         p          + 5759p         p         p        |
|      1, 1} {0, 2, 0} {1, 2, 0} {1, 2, 1}        {0, 1, 1} {1, 0, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                           2                                                 |
|        p          + 10459p         p         p         p          - 7355p   |
|      0} {1, 2, 1}         {0, 2, 0} {1, 0, 0} {1, 2, 0} {1, 2, 1}        {0,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p         p          - 15256p         p         p       |
|      1, 0} {1, 0, 0} {1, 2, 0} {1, 2, 1}         {0, 1, 1} {1, 0, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                    2                                     3  |
|        p          + 1473p         p         p         p          + 12431p   |
|      0} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 2, 0} {1, 2, 1}         {1,|
|      -----------------------------------------------------------------------|
|                                      2                                      |
|           p         p          - 553p         p         p         p         |
|      0, 0} {1, 2, 0} {1, 2, 1}       {0, 0, 1} {1, 0, 1} {1, 2, 0} {1, 2, 1}|
|      -----------------------------------------------------------------------|
|                                                                             |
|      + 12541p         p         p         p         p          + 6208p      |
|              {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 2, 0} {1, 2, 1}        {0, 0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          - 5368p         p           |
|      0} {0, 1, 1} {1, 0, 1} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {0, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 6936p         p         p           |
|      1} {1, 0, 1} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {0, 1, 1} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          + 3524p         p         p         p          +|
|      1} {1, 2, 0} {1, 2, 1}        {0, 1, 1} {1, 0, 1} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      14650p         p         p         p         p          + 4986p        |
|            {0, 0, 1} {0, 2, 0} {1, 0, 1} {1, 2, 0} {1, 2, 1}        {0, 1,  |
|      -----------------------------------------------------------------------|
|                                                         2                   |
|        p         p         p         p          + 12250p         p          |
|      1} {0, 2, 0} {1, 0, 1} {1, 2, 0} {1, 2, 1}         {0, 2, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 5735p         p         p         p           |
|      1} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 2,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 1566p         p         p         p         p          +|
|      0} {1, 2, 1}        {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                     2       |
|      13649p         p         p         p         p          - 4472p        |
|            {0, 2, 0} {1, 0, 0} {1, 0, 1} {1, 2, 0} {1, 2, 1}        {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                         2                   |
|        p         p         p          - 11246p         p         p          |
|      0} {1, 0, 1} {1, 2, 0} {1, 2, 1}         {0, 0, 0} {1, 0, 1} {1, 2,    |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          - 10469p         p         p         p          + 2008p   |
|      0} {1, 2, 1}         {0, 0, 1} {1, 0, 1} {1, 2, 0} {1, 2, 1}        {0,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p         p          - 10278p         p         p       |
|      1, 0} {1, 0, 1} {1, 2, 0} {1, 2, 1}         {0, 1, 1} {1, 0, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                                     2                                       |
|        p          - 14332p         p         p         p          + 2500p   |
|      0} {1, 2, 1}         {0, 2, 0} {1, 0, 1} {1, 2, 0} {1, 2, 1}        {1,|
|      -----------------------------------------------------------------------|
|            2                                     3                          |
|           p         p         p          + 10862p         p         p       |
|      0, 0} {1, 0, 1} {1, 2, 0} {1, 2, 1}         {1, 0, 1} {1, 2, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                 2                                                           |
|         - 11982p         p         p         p          - 11147p            |
|      1}         {0, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}         {0, 0,      |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p         p          - 5983p         p           |
|      1} {0, 1, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 15906p         p         p         p          |
|      0} {1, 2, 0} {1, 2, 1}         {0, 0, 1} {0, 2, 0} {1, 1, 0} {1, 2,    |
|      -----------------------------------------------------------------------|
|                           2                                                 |
|        p          + 13284p         p         p         p          - 2160p   |
|      0} {1, 2, 1}         {0, 2, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p         p          + 11145p         p       |
|      1, 0} {1, 0, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}         {0, 2, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p         p          + 1269p         p         p           |
|      0} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {1, 0, 0} {1, 1, 0} {1, 2,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 11000p         p         p         p         p          |
|      0} {1, 2, 1}         {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                                                             |
|      - 12144p         p         p         p         p          + 5179p      |
|              {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p         p          - 5139p         p           |
|      0} {1, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, 2, 0} {1, 0,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 14440p         p         p          |
|      1} {1, 1, 0} {1, 2, 0} {1, 2, 1}         {1, 0, 0} {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          + 1351p         p         p         p          -|
|      0} {1, 2, 0} {1, 2, 1}        {1, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                              2        |
|      7003p         p         p         p          - 7775p         p         |
|           {0, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {1, 1,   |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          + 6033p         p         p         p          +|
|      0} {1, 2, 0} {1, 2, 1}        {0, 2, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                             2         |
|      9403p         p         p         p          + 374p         p          |
|           {1, 0, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}       {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                    3                                    2   |
|        p         p          - 7742p         p         p          - 9188p    |
|      0} {1, 2, 0} {1, 2, 1}        {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p          + 4024p         p         p        |
|      0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|        p         p          + 7926p         p         p         p          +|
|      1} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      5004p         p         p         p         p          + 1390p         |
|           {0, 0, 1} {0, 1, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 1,   |
|      -----------------------------------------------------------------------|
|                                                         2                   |
|        p         p         p         p          - 10836p         p          |
|      0} {0, 1, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 1, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 10549p         p         p         p          |
|      1} {1, 2, 0} {1, 2, 1}         {0, 0, 1} {0, 2, 0} {1, 1, 1} {1, 2,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 298p         p         p         p         p          - |
|      0} {1, 2, 1}       {0, 1, 1} {0, 2, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}   |
|      -----------------------------------------------------------------------|
|           2                                                                 |
|      7784p         p         p         p          + 3920p         p         |
|           {0, 2, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {1, 0,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 15084p         p         p          |
|      0} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 1, 1} {1, 0, 0} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          + 8687p         p         p         p           |
|      1} {1, 2, 0} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 1, 1} {1, 2,     |
|      -----------------------------------------------------------------------|
|                           2                                                 |
|        p          + 14306p         p         p         p          - 4467p   |
|      0} {1, 2, 1}         {1, 0, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0,|
|      -----------------------------------------------------------------------|
|                                                                             |
|           p         p         p         p          - 2704p         p        |
|      0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          - 15456p         p         p          |
|      1} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 1, 0} {1, 0, 1} {1, 1,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 13244p         p         p         p          |
|      1} {1, 2, 0} {1, 2, 1}         {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 294p         p         p         p         p          + |
|      0} {1, 2, 1}       {0, 2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}   |
|      -----------------------------------------------------------------------|
|                                                                     2       |
|      11752p         p         p         p         p          - 3763p        |
|            {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p         p          + 8464p         p         p           |
|      1} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {1, 1, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p         p          - 2826p         p         p         p           |
|      1} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2,     |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 7074p         p         p         p         p          +|
|      0} {1, 2, 1}        {0, 2, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      1673p         p         p         p         p          - 11193p        |
|           {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {1, 0,  |
|      -----------------------------------------------------------------------|
|                                                        2                    |
|        p         p         p         p          + 9946p         p           |
|      1} {1, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {1, 1, 0} {1, 1,     |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          + 3902p         p         p         p          +|
|      1} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                     2                                              2        |
|      8241p         p         p         p          - 6596p         p         |
|           {0, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 1, 1} {1, 1,   |
|      -----------------------------------------------------------------------|
|                                               2                             |
|        p         p          + 15216p         p         p         p          |
|      1} {1, 2, 0} {1, 2, 1}         {0, 2, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                       2                                               2     |
|      + 3863p         p         p         p          - 12521p         p      |
|             {1, 0, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {1, 0, 1} {1, 1,|
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p         p          + 5887p         p         p         p          -|
|      1} {1, 2, 0} {1, 2, 1}        {1, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|            3                                     2         2                |
|      12028p         p         p          - 10179p         p         p       |
|            {1, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 0, 1} {1, 2, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         - 5491p         p         p         p          + 10588p         p   |
|      1}        {0, 0, 1} {0, 1, 0} {1, 2, 0} {1, 2, 1}         {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p         p          + 6339p         p         p         p        |
|      1, 1} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {0, 1, 1} {1, 2, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                2         2                                               2  |
|         + 4682p         p         p          - 10454p         p         p   |
|      1}        {0, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 0, 1} {0, 2, 0} {1,|
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p          - 6363p         p         p         p          +       |
|      2, 0} {1, 2, 1}        {0, 1, 1} {0, 2, 0} {1, 2, 0} {1, 2, 1}         |
|      -----------------------------------------------------------------------|
|           2         2                                               2       |
|      9514p         p         p          - 12679p         p         p        |
|           {0, 2, 0} {1, 2, 0} {1, 2, 1}         {0, 1, 0} {1, 0, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          - 6532p         p         p         p          - 6208p    |
|      0} {1, 2, 1}        {0, 1, 1} {1, 0, 0} {1, 2, 0} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                      2                           2         2                |
|           p         p         p          + 15795p         p         p       |
|      2, 0} {1, 0, 0} {1, 2, 0} {1, 2, 1}         {1, 0, 0} {1, 2, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         + 5065p         p         p         p          - 6848p         p    |
|      1}        {0, 0, 0} {1, 0, 1} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          - 14887p         p         p         p       |
|      0, 1} {1, 2, 0} {1, 2, 1}         {0, 1, 0} {1, 0, 1} {1, 2, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                    2                                        |
|         - 9633p         p         p         p          - 4441p         p    |
|      1}        {0, 1, 1} {1, 0, 1} {1, 2, 0} {1, 2, 1}        {0, 2, 0} {1, |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p         p          + 15166p         p         p         p       |
|      0, 1} {1, 2, 0} {1, 2, 1}         {1, 0, 0} {1, 0, 1} {1, 2, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                2         2                                              2   |
|         + 9786p         p         p          + 5485p         p         p    |
|      1}        {1, 0, 1} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {1, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|                                                 2                           |
|           p          - 4589p         p         p         p          -       |
|      2, 0} {1, 2, 1}        {0, 1, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}         |
|      -----------------------------------------------------------------------|
|                                2                                            |
|      15163p         p         p         p          + 12437p         p       |
|            {0, 2, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}         {1, 0, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p         p          - 10679p         p         p         p          |
|      0} {1, 2, 0} {1, 2, 1}         {1, 0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|             2         2                                              2      |
|      + 3194p         p         p          + 7352p         p         p       |
|             {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                                             2                               |
|        p          - 754p         p         p         p          - 11016p    |
|      0} {1, 2, 1}       {0, 1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {0, |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p         p          + 13170p         p         p       |
|      1, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 2, 0} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|        p          + 6957p         p         p         p          - 9884p    |
|      0} {1, 2, 1}        {1, 0, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {1, |
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p         p          + 11406p         p         p       |
|      0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {1, 1, 0} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|                          2         2                                    3   |
|        p          - 8050p         p         p          - 4505p         p    |
|      0} {1, 2, 1}        {1, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|                                       3                                     |
|           p          + 6366p         p         p          - 7472p           |
|      2, 0} {1, 2, 1}        {0, 1, 0} {1, 2, 0} {1, 2, 1}        {0, 1,     |
|      -----------------------------------------------------------------------|
|         3                                    3                              |
|        p         p          + 4382p         p         p          + 1408p    |
|      1} {1, 2, 0} {1, 2, 1}        {0, 2, 0} {1, 2, 0} {1, 2, 1}        {1, |
|      -----------------------------------------------------------------------|
|            3                                    3                           |
|           p         p          + 5784p         p         p          -       |
|      0, 0} {1, 2, 0} {1, 2, 1}        {1, 0, 1} {1, 2, 0} {1, 2, 1}         |
|      -----------------------------------------------------------------------|
|                     3                                     3                 |
|      9259p         p         p          - 10182p         p         p        |
|           {1, 1, 0} {1, 2, 0} {1, 2, 1}         {1, 1, 1} {1, 2, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                 2                   2                                       |
|         - 11502p         p         p          - 9252p         p         p   |
|      1}         {0, 0, 1} {0, 1, 1} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {0,|
|      -----------------------------------------------------------------------|
|            2                          2         2                           |
|           p          + 5803p         p         p          - 7044p           |
|      1, 1} {1, 2, 1}        {0, 0, 1} {0, 1, 1} {1, 2, 1}        {0, 1,     |
|      -----------------------------------------------------------------------|
|         2         2                 3         2                             |
|        p         p          + 15226p         p          - 3530p         p   |
|      0} {0, 1, 1} {1, 2, 1}         {0, 1, 1} {1, 2, 1}        {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|                      2                 2                   2                |
|           p         p          - 10994p         p         p          +      |
|      1, 1} {0, 2, 0} {1, 2, 1}         {0, 1, 1} {0, 2, 0} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                      2         2                          2         2       |
|      11945p         p         p          - 7907p         p         p        |
|            {0, 0, 1} {0, 2, 0} {1, 2, 1}        {0, 1, 1} {0, 2, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                 3         2                 2                   2           |
|         + 12114p         p          + 10467p         p         p          - |
|      1}         {0, 2, 0} {1, 2, 1}         {0, 0, 1} {0, 2, 1} {1, 2, 1}   |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      15333p         p         p         p          - 3102p         p        |
|            {0, 0, 1} {0, 1, 0} {0, 2, 1} {1, 2, 1}        {0, 0, 1} {0, 1,  |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p          - 4056p         p         p         p          -|
|      1} {0, 2, 1} {1, 2, 1}        {0, 1, 0} {0, 1, 1} {0, 2, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|           2                   2                                             |
|      3434p         p         p          - 2558p         p         p         |
|           {0, 1, 1} {0, 2, 1} {1, 2, 1}        {0, 0, 1} {0, 2, 0} {0, 2,   |
|      -----------------------------------------------------------------------|
|         2                                              2                 2  |
|        p          + 5643p         p         p         p          + 11467p   |
|      1} {1, 2, 1}        {0, 1, 1} {0, 2, 0} {0, 2, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|                      2                          2         2                 |
|           p         p          - 9566p         p         p          +       |
|      2, 0} {0, 2, 1} {1, 2, 1}        {0, 0, 1} {0, 2, 1} {1, 2, 1}         |
|      -----------------------------------------------------------------------|
|                      2         2                          2         2       |
|      14270p         p         p          + 7574p         p         p        |
|            {0, 1, 0} {0, 2, 1} {1, 2, 1}        {0, 1, 1} {0, 2, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                           2         2                3         2            |
|         - 13321p         p         p          + 6739p         p          +  |
|      1}         {0, 2, 0} {0, 2, 1} {1, 2, 1}        {0, 2, 1} {1, 2, 1}    |
|      -----------------------------------------------------------------------|
|            2                   2                2                   2       |
|      12196p         p         p          + 3967p         p         p        |
|            {0, 1, 1} {1, 0, 0} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|         + 3219p         p         p         p          + 7074p         p    |
|      1}        {0, 1, 0} {0, 2, 1} {1, 0, 0} {1, 2, 1}        {0, 1, 1} {0, |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          - 5177p         p         p         p        |
|      2, 1} {1, 0, 0} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 0, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                2                   2                          2         2   |
|         - 2333p         p         p          - 5162p         p         p    |
|      1}        {0, 2, 1} {1, 0, 0} {1, 2, 1}        {0, 1, 0} {1, 0, 0} {1, |
|      -----------------------------------------------------------------------|
|                             2         2                           2         |
|            - 9602p         p         p          + 12675p         p          |
|      2, 1}        {0, 1, 1} {1, 0, 0} {1, 2, 1}         {0, 2, 0} {1, 0,    |
|      -----------------------------------------------------------------------|
|         2                         2         2                3         2    |
|        p          + 173p         p         p          + 9754p         p     |
|      0} {1, 2, 1}       {0, 2, 1} {1, 0, 0} {1, 2, 1}        {1, 0, 0} {1,  |
|      -----------------------------------------------------------------------|
|                   2                   2                                     |
|            - 6929p         p         p          - 11046p         p          |
|      2, 1}        {0, 0, 1} {1, 0, 1} {1, 2, 1}         {0, 0, 1} {0, 1,    |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p          + 4704p         p         p         p          -|
|      0} {1, 0, 1} {1, 2, 1}        {0, 0, 0} {0, 1, 1} {1, 0, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                         2                                   |
|      4969p         p         p         p          - 6726p         p         |
|           {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 2, 1}        {0, 1, 0} {0, 1,   |
|      -----------------------------------------------------------------------|
|                   2                 2                   2                   |
|        p         p          - 10136p         p         p          + 9606p   |
|      1} {1, 0, 1} {1, 2, 1}         {0, 1, 1} {1, 0, 1} {1, 2, 1}        {0,|
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          - 14027p         p         p       |
|      0, 1} {0, 2, 0} {1, 0, 1} {1, 2, 1}         {0, 1, 1} {0, 2, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|         2                2                   2                              |
|        p          - 1732p         p         p          - 11186p         p   |
|      1} {1, 2, 1}        {0, 2, 0} {1, 0, 1} {1, 2, 1}         {0, 0, 0} {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          + 2662p         p         p         p        |
|      2, 1} {1, 0, 1} {1, 2, 1}        {0, 0, 1} {0, 2, 1} {1, 0, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                              2                              |
|         + 5399p         p         p         p          + 13029p         p   |
|      1}        {0, 1, 0} {0, 2, 1} {1, 0, 1} {1, 2, 1}         {0, 1, 1} {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          - 5334p         p         p         p        |
|      2, 1} {1, 0, 1} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 0, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|               2                   2                                         |
|         + 124p         p         p          + 8152p         p         p     |
|      1}       {0, 2, 1} {1, 0, 1} {1, 2, 1}        {0, 1, 0} {1, 0, 0} {1,  |
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p          - 11006p         p         p         p          -      |
|      0, 1} {1, 2, 1}         {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                                         2                                   |
|      9447p         p         p         p          + 982p         p          |
|           {0, 2, 0} {1, 0, 0} {1, 0, 1} {1, 2, 1}       {0, 2, 1} {1, 0,    |
|      -----------------------------------------------------------------------|
|                   2                2                   2                    |
|        p         p          - 2439p         p         p          - 14500p   |
|      0} {1, 0, 1} {1, 2, 1}        {1, 0, 0} {1, 0, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|            2         2                           2         2                |
|           p         p          - 10811p         p         p          -      |
|      0, 0} {1, 0, 1} {1, 2, 1}         {0, 0, 1} {1, 0, 1} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                     2         2                          2         2        |
|      2016p         p         p          + 3271p         p         p         |
|           {0, 1, 0} {1, 0, 1} {1, 2, 1}        {0, 1, 1} {1, 0, 1} {1, 2, 1}|
|      -----------------------------------------------------------------------|
|                        2         2                           2         2    |
|      + 13216p         p         p          + 12366p         p         p     |
|              {0, 2, 0} {1, 0, 1} {1, 2, 1}         {0, 2, 1} {1, 0, 1} {1,  |
|      -----------------------------------------------------------------------|
|                             2         2                 3         2         |
|            + 8055p         p         p          - 11008p         p          |
|      2, 1}        {1, 0, 0} {1, 0, 1} {1, 2, 1}         {1, 0, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|              2                   2                                          |
|      - 13581p         p         p          + 6216p         p         p      |
|              {0, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {1, 1,|
|      -----------------------------------------------------------------------|
|         2                 2                   2                             |
|        p          - 13368p         p         p          - 7596p         p   |
|      0} {1, 2, 1}         {0, 1, 0} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|                      2                 2                   2                |
|           p         p          + 13137p         p         p          +      |
|      2, 0} {1, 1, 0} {1, 2, 1}         {0, 2, 0} {1, 1, 0} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      10248p         p         p         p          - 7299p         p        |
|            {0, 0, 1} {0, 2, 1} {1, 1, 0} {1, 2, 1}        {0, 1, 0} {0, 2,  |
|      -----------------------------------------------------------------------|
|                   2                                              2          |
|        p         p          + 9834p         p         p         p          -|
|      1} {1, 1, 0} {1, 2, 1}        {0, 2, 0} {0, 2, 1} {1, 1, 0} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|           2                   2                                             |
|      1748p         p         p          - 14882p         p         p        |
|           {0, 2, 1} {1, 1, 0} {1, 2, 1}         {0, 1, 0} {1, 0, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|         2                                             2                     |
|        p          + 386p         p         p         p          + 8309p     |
|      0} {1, 2, 1}       {0, 2, 0} {1, 0, 0} {1, 1, 0} {1, 2, 1}        {0,  |
|      -----------------------------------------------------------------------|
|                                2                 2                   2      |
|           p         p         p          + 15874p         p         p       |
|      2, 1} {1, 0, 0} {1, 1, 0} {1, 2, 1}         {1, 0, 0} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|         + 4079p         p         p         p          + 11324p         p   |
|      1}        {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}         {0, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                      2                                               2      |
|           p         p          - 10677p         p         p         p       |
|      0, 1} {1, 1, 0} {1, 2, 1}         {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                              2                              |
|         - 7899p         p         p         p          - 14715p         p   |
|      1}        {0, 2, 0} {1, 0, 1} {1, 1, 0} {1, 2, 1}         {0, 2, 1} {1,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          - 7938p         p         p         p        |
|      0, 1} {1, 1, 0} {1, 2, 1}        {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|               2                   2                          2         2    |
|         + 679p         p         p          + 6484p         p         p     |
|      1}       {1, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 0, 1} {1, 1, 0} {1,  |
|      -----------------------------------------------------------------------|
|                              2         2                           2        |
|            - 14561p         p         p          - 14451p         p         |
|      2, 1}         {0, 1, 0} {1, 1, 0} {1, 2, 1}         {0, 2, 0} {1, 1,   |
|      -----------------------------------------------------------------------|
|         2                           2         2                          2  |
|        p          + 11089p         p         p          - 1769p         p   |
|      0} {1, 2, 1}         {0, 2, 1} {1, 1, 0} {1, 2, 1}        {1, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|            2                           2         2                 3        |
|           p          - 11959p         p         p          - 12842p         |
|      1, 0} {1, 2, 1}         {1, 0, 1} {1, 1, 0} {1, 2, 1}         {1, 1,   |
|      -----------------------------------------------------------------------|
|         2                 2                   2                             |
|        p          - 10801p         p         p          - 4104p         p   |
|      0} {1, 2, 1}         {0, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|                      2                 2                   2                |
|           p         p          - 15346p         p         p          -      |
|      1, 0} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, 1, 1} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                                        2                                    |
|      421p         p         p         p          - 6515p         p          |
|          {0, 0, 1} {0, 1, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 0} {0, 1,    |
|      -----------------------------------------------------------------------|
|                   2                2                   2                    |
|        p         p          + 5479p         p         p          - 3394p    |
|      1} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {1, 1, 1} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          + 1678p         p         p        |
|      0, 1} {0, 2, 0} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {0, 2, 0} {1, 1,  |
|      -----------------------------------------------------------------------|
|         2                 2                   2                             |
|        p          + 13300p         p         p          + 8014p         p   |
|      1} {1, 2, 1}         {0, 2, 0} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {0,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          + 8235p         p         p         p        |
|      2, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 0} {0, 2, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                               2                             |
|         + 10465p         p         p         p          - 13479p            |
|      1}         {0, 1, 1} {0, 2, 1} {1, 1, 1} {1, 2, 1}         {0, 2,      |
|      -----------------------------------------------------------------------|
|                             2               2                   2           |
|        p         p         p          - 574p         p         p          - |
|      0} {0, 2, 1} {1, 1, 1} {1, 2, 1}       {0, 2, 1} {1, 1, 1} {1, 2, 1}   |
|      -----------------------------------------------------------------------|
|                                         2                                   |
|      3372p         p         p         p          + 1359p         p         |
|           {0, 1, 0} {1, 0, 0} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {1, 0,   |
|      -----------------------------------------------------------------------|
|                   2                                               2         |
|        p         p          - 11030p         p         p         p          |
|      0} {1, 1, 1} {1, 2, 1}         {0, 2, 0} {1, 0, 0} {1, 1, 1} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                           2                 2               |
|      - 2887p         p         p         p          + 13070p         p      |
|             {0, 2, 1} {1, 0, 0} {1, 1, 1} {1, 2, 1}         {1, 0, 0} {1, 1,|
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p          + 9885p         p         p         p          - 11656p   |
|      1} {1, 2, 1}        {0, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          - 12582p         p         p       |
|      0, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0, 1, 0} {1, 0, 1} {1, 1, |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p          + 4189p         p         p         p          + 11800p   |
|      1} {1, 2, 1}        {0, 1, 1} {1, 0, 1} {1, 1, 1} {1, 2, 1}         {0,|
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          + 6138p         p         p        |
|      2, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|         2                                               2                2  |
|        p          + 15086p         p         p         p          - 3017p   |
|      1} {1, 2, 1}         {1, 0, 0} {1, 0, 1} {1, 1, 1} {1, 2, 1}        {1,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          - 8850p         p         p         p        |
|      0, 1} {1, 1, 1} {1, 2, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                               2                             |
|         + 15746p         p         p         p          + 9909p         p   |
|      1}         {0, 1, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 0} {1,|
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          + 1607p         p         p         p        |
|      1, 0} {1, 1, 1} {1, 2, 1}        {0, 2, 1} {1, 1, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                               2                             |
|         - 11100p         p         p         p          - 1055p         p   |
|      1}         {1, 0, 0} {1, 1, 0} {1, 1, 1} {1, 2, 1}        {1, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|                      2                 2                   2                |
|           p         p          + 14015p         p         p          +      |
|      1, 0} {1, 1, 1} {1, 2, 1}         {1, 1, 0} {1, 1, 1} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                      2         2                          2         2       |
|      15897p         p         p          - 6969p         p         p        |
|            {0, 0, 1} {1, 1, 1} {1, 2, 1}        {0, 1, 0} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                          2         2                           2         2  |
|         + 6467p         p         p          - 12520p         p         p   |
|      1}        {0, 1, 1} {1, 1, 1} {1, 2, 1}         {0, 2, 0} {1, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|                              2         2                           2        |
|            + 12193p         p         p          + 11205p         p         |
|      2, 1}         {0, 2, 1} {1, 1, 1} {1, 2, 1}         {1, 0, 0} {1, 1,   |
|      -----------------------------------------------------------------------|
|         2                           2         2                             |
|        p          + 15156p         p         p          - 10224p            |
|      1} {1, 2, 1}         {1, 0, 1} {1, 1, 1} {1, 2, 1}         {1, 1,      |
|      -----------------------------------------------------------------------|
|         2         2                 3         2                2            |
|        p         p          - 11980p         p          - 9030p         p   |
|      0} {1, 1, 1} {1, 2, 1}         {1, 1, 1} {1, 2, 1}        {0, 0, 1} {1,|
|      -----------------------------------------------------------------------|
|            2                                              2                 |
|           p          - 3344p         p         p         p          +       |
|      2, 0} {1, 2, 1}        {0, 0, 1} {0, 1, 0} {1, 2, 0} {1, 2, 1}         |
|      -----------------------------------------------------------------------|
|                                         2                                   |
|      8552p         p         p         p          - 975p         p          |
|           {0, 0, 1} {0, 1, 1} {1, 2, 0} {1, 2, 1}       {0, 1, 0} {0, 1,    |
|      -----------------------------------------------------------------------|
|                   2                2                   2                    |
|        p         p          - 2364p         p         p          - 4020p    |
|      1} {1, 2, 0} {1, 2, 1}        {0, 1, 1} {1, 2, 0} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          - 667p         p         p         |
|      0, 1} {0, 2, 0} {1, 2, 0} {1, 2, 1}       {0, 1, 1} {0, 2, 0} {1, 2,   |
|      -----------------------------------------------------------------------|
|         2                2                   2                              |
|        p          - 6704p         p         p          - 3144p         p    |
|      0} {1, 2, 1}        {0, 2, 0} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|                      2                                              2       |
|           p         p          - 3628p         p         p         p        |
|      0, 0} {1, 2, 0} {1, 2, 1}        {0, 1, 1} {1, 0, 0} {1, 2, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                               2                2            |
|         + 12012p         p         p         p          - 7126p         p   |
|      1}         {0, 2, 0} {1, 0, 0} {1, 2, 0} {1, 2, 1}        {1, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|            2                                               2                |
|           p          - 15661p         p         p         p          +      |
|      2, 0} {1, 2, 1}         {0, 0, 0} {1, 0, 1} {1, 2, 0} {1, 2, 1}        |
|      -----------------------------------------------------------------------|
|                                         2                                   |
|      7424p         p         p         p          + 2004p         p         |
|           {0, 0, 1} {1, 0, 1} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {1, 0,   |
|      -----------------------------------------------------------------------|
|                   2                                               2         |
|        p         p          + 13814p         p         p         p          |
|      1} {1, 2, 0} {1, 2, 1}         {0, 1, 1} {1, 0, 1} {1, 2, 0} {1, 2, 1} |
|      -----------------------------------------------------------------------|
|                                           2                                 |
|      + 3299p         p         p         p          + 5690p         p       |
|             {0, 2, 0} {1, 0, 1} {1, 2, 0} {1, 2, 1}        {1, 0, 0} {1, 0, |
|      -----------------------------------------------------------------------|
|                   2               2                   2                     |
|        p         p          - 774p         p         p          - 12287p    |
|      1} {1, 2, 0} {1, 2, 1}       {1, 0, 1} {1, 2, 0} {1, 2, 1}         {0, |
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          - 1989p         p         p        |
|      0, 1} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, 1, 0} {1, 1, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|         2                                               2                   |
|        p          + 12287p         p         p         p          -         |
|      0} {1, 2, 1}         {0, 2, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}           |
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      11494p         p         p         p          - 7463p         p        |
|            {1, 0, 0} {1, 1, 0} {1, 2, 0} {1, 2, 1}        {1, 0, 1} {1, 1,  |
|      -----------------------------------------------------------------------|
|                   2                2                   2                    |
|        p         p          - 3656p         p         p          + 8658p    |
|      0} {1, 2, 0} {1, 2, 1}        {1, 1, 0} {1, 2, 0} {1, 2, 1}        {0, |
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          + 10134p         p         p       |
|      0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 1, 0} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p          + 8552p         p         p         p          + 795p     |
|      0} {1, 2, 1}        {0, 1, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}       {0,  |
|      -----------------------------------------------------------------------|
|                                2                                            |
|           p         p         p          + 11697p         p         p       |
|      2, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {1, 0, 0} {1, 1, 1} {1, 2, |
|      -----------------------------------------------------------------------|
|         2                                              2                    |
|        p          - 8219p         p         p         p          + 13338p   |
|      0} {1, 2, 1}        {1, 0, 1} {1, 1, 1} {1, 2, 0} {1, 2, 1}         {1,|
|      -----------------------------------------------------------------------|
|                                2                2                   2       |
|           p         p         p          - 6641p         p         p        |
|      1, 0} {1, 1, 1} {1, 2, 0} {1, 2, 1}        {1, 1, 1} {1, 2, 0} {1, 2,  |
|      -----------------------------------------------------------------------|
|                          2         2                           2         2  |
|         + 9541p         p         p          + 15344p         p         p   |
|      1}        {0, 0, 1} {1, 2, 0} {1, 2, 1}         {0, 1, 0} {1, 2, 0} {1,|
|      -----------------------------------------------------------------------|
|                              2         2                           2        |
|            - 10369p         p         p          + 12025p         p         |
|      2, 1}         {0, 1, 1} {1, 2, 0} {1, 2, 1}         {0, 2, 0} {1, 2,   |
|      -----------------------------------------------------------------------|
|         2                          2         2                          2   |
|        p          + 8531p         p         p          + 5888p         p    |
|      0} {1, 2, 1}        {1, 0, 0} {1, 2, 0} {1, 2, 1}        {1, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|            2                          2         2                           |
|           p          - 3407p         p         p          - 5964p           |
|      2, 0} {1, 2, 1}        {1, 1, 0} {1, 2, 0} {1, 2, 1}        {1, 1,     |
|      -----------------------------------------------------------------------|
|         2         2                3         2                2         3   |
|        p         p          + 1466p         p          + 8880p         p    |
|      1} {1, 2, 0} {1, 2, 1}        {1, 2, 0} {1, 2, 1}        {0, 0, 1} {1, |
|      -----------------------------------------------------------------------|
|                                       3                                     |
|            + 8233p         p         p          + 1538p         p           |
|      2, 1}        {0, 0, 1} {0, 1, 0} {1, 2, 1}        {0, 0, 1} {0, 1,     |
|      -----------------------------------------------------------------------|
|         3                                    3                 2         3  |
|        p          + 1938p         p         p          - 14866p         p   |
|      1} {1, 2, 1}        {0, 1, 0} {0, 1, 1} {1, 2, 1}         {0, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|                                       3                                     |
|            + 4699p         p         p          - 2416p         p           |
|      2, 1}        {0, 0, 1} {0, 2, 0} {1, 2, 1}        {0, 1, 1} {0, 2,     |
|      -----------------------------------------------------------------------|
|         3                2         3                                   3    |
|        p          + 4918p         p          - 141p         p         p     |
|      0} {1, 2, 1}        {0, 2, 0} {1, 2, 1}       {0, 0, 1} {0, 2, 1} {1,  |
|      -----------------------------------------------------------------------|
|                                      3                                      |
|            - 835p         p         p          - 1889p         p            |
|      2, 1}       {0, 1, 0} {0, 2, 1} {1, 2, 1}        {0, 1, 1} {0, 2,      |
|      -----------------------------------------------------------------------|
|         3                                     3                 2           |
|        p          - 11070p         p         p          - 14646p            |
|      1} {1, 2, 1}         {0, 2, 0} {0, 2, 1} {1, 2, 1}         {0, 2,      |
|      -----------------------------------------------------------------------|
|         3                                    3                              |
|        p          + 6507p         p         p          + 13073p         p   |
|      1} {1, 2, 1}        {0, 1, 0} {1, 0, 0} {1, 2, 1}         {0, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|            3                                    3                           |
|           p          - 8853p         p         p          - 12794p          |
|      0, 0} {1, 2, 1}        {0, 2, 0} {1, 0, 0} {1, 2, 1}         {0, 2,    |
|      -----------------------------------------------------------------------|
|                   3                2         3                              |
|        p         p          - 7802p         p          + 15642p         p   |
|      1} {1, 0, 0} {1, 2, 1}        {1, 0, 0} {1, 2, 1}         {0, 0, 0} {1,|
|      -----------------------------------------------------------------------|
|            3                                     3                          |
|           p          - 12610p         p         p          + 10538p         |
|      0, 1} {1, 2, 1}         {0, 0, 1} {1, 0, 1} {1, 2, 1}         {0, 1,   |
|      -----------------------------------------------------------------------|
|                   3                                     3                   |
|        p         p          + 14212p         p         p          -         |
|      0} {1, 0, 1} {1, 2, 1}         {0, 1, 1} {1, 0, 1} {1, 2, 1}           |
|      -----------------------------------------------------------------------|
|                                3                                    3       |
|      15806p         p         p          - 4090p         p         p        |
|            {0, 2, 0} {1, 0, 1} {1, 2, 1}        {0, 2, 1} {1, 0, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                     3                2         3            |
|         + 13197p         p         p          + 5683p         p          -  |
|      1}         {1, 0, 0} {1, 0, 1} {1, 2, 1}        {1, 0, 1} {1, 2, 1}    |
|      -----------------------------------------------------------------------|
|                               3                                    3        |
|      2748p         p         p          + 6449p         p         p         |
|           {0, 0, 1} {1, 1, 0} {1, 2, 1}        {0, 1, 0} {1, 1, 0} {1, 2, 1}|
|      -----------------------------------------------------------------------|
|                                 3                                    3      |
|      - 8634p         p         p          + 1928p         p         p       |
|             {0, 2, 0} {1, 1, 0} {1, 2, 1}        {0, 2, 1} {1, 1, 0} {1, 2, |
|      -----------------------------------------------------------------------|
|                                     3                                    3  |
|         - 15647p         p         p          - 5641p         p         p   |
|      1}         {1, 0, 0} {1, 1, 0} {1, 2, 1}        {1, 0, 1} {1, 1, 0} {1,|
|      -----------------------------------------------------------------------|
|                   2         3                                    3          |
|            - 2364p         p          + 3351p         p         p          -|
|      2, 1}        {1, 1, 0} {1, 2, 1}        {0, 0, 1} {1, 1, 1} {1, 2, 1}  |
|      -----------------------------------------------------------------------|
|                                3                                    3       |
|      14770p         p         p          - 2535p         p         p        |
|            {0, 1, 0} {1, 1, 1} {1, 2, 1}        {0, 1, 1} {1, 1, 1} {1, 2,  |
|      -----------------------------------------------------------------------|
|                                    3                                     3  |
|         + 1287p         p         p          + 10326p         p         p   |
|      1}        {0, 2, 0} {1, 1, 1} {1, 2, 1}         {0, 2, 1} {1, 1, 1} {1,|
|      -----------------------------------------------------------------------|
|                                       3                                     |
|            - 3562p         p         p          + 8769p         p           |
|      2, 1}        {1, 0, 0} {1, 1, 1} {1, 2, 1}        {1, 0, 1} {1, 1,     |
|      -----------------------------------------------------------------------|
|         3                                   3                2         3    |
|        p          - 335p         p         p          + 4391p         p     |
|      1} {1, 2, 1}       {1, 1, 0} {1, 1, 1} {1, 2, 1}        {1, 1, 1} {1,  |
|      -----------------------------------------------------------------------|
|                                       3                                     |
|            - 1466p         p         p          - 10782p         p          |
|      2, 1}        {0, 0, 1} {1, 2, 0} {1, 2, 1}         {0, 1, 0} {1, 2,    |
|      -----------------------------------------------------------------------|
|         3                                    3                              |
|        p          - 7130p         p         p          - 7017p         p    |
|      0} {1, 2, 1}        {0, 1, 1} {1, 2, 0} {1, 2, 1}        {0, 2, 0} {1, |
|      -----------------------------------------------------------------------|
|            3                                    3                           |
|           p          + 2317p         p         p          + 12664p          |
|      2, 0} {1, 2, 1}        {1, 0, 0} {1, 2, 0} {1, 2, 1}         {1, 0,    |
|      -----------------------------------------------------------------------|
|                   3                                     3                   |
|        p         p          - 12994p         p         p          - 7391p   |
|      1} {1, 2, 0} {1, 2, 1}         {1, 1, 0} {1, 2, 0} {1, 2, 1}        {1,|
|      -----------------------------------------------------------------------|
|                      3                2         3                           |
|           p         p          - 4282p         p          + 14251p          |
|      1, 1} {1, 2, 0} {1, 2, 1}        {1, 2, 0} {1, 2, 1}         {0, 0,    |
|      -----------------------------------------------------------------------|
|         4                          4                          4             |
|        p          + 5512p         p          + 1006p         p          -   |
|      1} {1, 2, 1}        {0, 1, 0} {1, 2, 1}        {0, 1, 1} {1, 2, 1}     |
|      -----------------------------------------------------------------------|
|                     4                           4                           |
|      2995p         p          - 14910p         p          - 13048p          |
|           {0, 2, 0} {1, 2, 1}         {0, 2, 1} {1, 2, 1}         {1, 0,    |
|      -----------------------------------------------------------------------|
|         4                          4                          4             |
|        p          - 6252p         p          + 9395p         p          +   |
|      0} {1, 2, 1}        {1, 0, 1} {1, 2, 1}        {1, 1, 0} {1, 2, 1}     |
|      -----------------------------------------------------------------------|
|                     4                          4                            |
|      5752p         p          + 4223p         p         )                   |
|           {1, 1, 1} {1, 2, 1}        {1, 2, 0} {1, 2, 1}                    |
|                                                                             |
|o15 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i16 : I2 = spohnCI(PR,X,G2, {a,b,c})                                         |
|                                                                             |
|o16 = ideal (p         p          - 15623p         p          - 15130p       |
|              {0, 0, 0} {1, 0, 0}         {0, 0, 1} {1, 0, 0}         {0, 1, |
|      -----------------------------------------------------------------------|
|        p          - 8126p         p          + 15935p         p          +  |
|      0} {1, 0, 0}        {0, 1, 1} {1, 0, 0}         {0, 2, 0} {1, 0, 0}    |
|      -----------------------------------------------------------------------|
|      1045p         p          + 479p         p          - 15145p            |
|           {0, 2, 1} {1, 0, 0}       {0, 0, 0} {1, 0, 1}         {0, 0,      |
|      -----------------------------------------------------------------------|
|        p          - 14652p         p          - 7648p         p          -  |
|      1} {1, 0, 1}         {0, 1, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}    |
|      -----------------------------------------------------------------------|
|      15590p         p          + 1523p         p          - 3924p           |
|            {0, 2, 0} {1, 0, 1}        {0, 2, 1} {1, 0, 1}        {0, 0,     |
|      -----------------------------------------------------------------------|
|        p          + 12455p         p          + 12948p         p          - |
|      0} {1, 1, 0}         {0, 0, 1} {1, 1, 0}         {0, 1, 0} {1, 1, 0}   |
|      -----------------------------------------------------------------------|
|      12051p         p          + 12010p         p          - 2880p          |
|            {0, 1, 1} {1, 1, 0}         {0, 2, 0} {1, 1, 0}        {0, 2,    |
|      -----------------------------------------------------------------------|
|        p          - 15478p         p          + 901p         p          +   |
|      1} {1, 1, 0}         {0, 0, 0} {1, 1, 1}       {0, 0, 1} {1, 1, 1}     |
|      -----------------------------------------------------------------------|
|      1394p         p          + 8398p         p          + 456p         p   |
|           {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}       {0, 2, 0} {1,|
|      -----------------------------------------------------------------------|
|            - 14434p         p          + 10874p         p          -        |
|      1, 1}         {0, 2, 1} {1, 1, 1}         {0, 0, 0} {1, 2, 0}          |
|      -----------------------------------------------------------------------|
|      4750p         p          - 4257p         p          + 2747p            |
|           {0, 0, 1} {1, 2, 0}        {0, 1, 0} {1, 2, 0}        {0, 1,      |
|      -----------------------------------------------------------------------|
|        p          - 5195p         p          + 11918p         p          -  |
|      1} {1, 2, 0}        {0, 2, 0} {1, 2, 0}         {0, 2, 1} {1, 2, 0}    |
|      -----------------------------------------------------------------------|
|      1623p         p          + 14756p         p          + 15249p          |
|           {0, 0, 0} {1, 2, 1}         {0, 0, 1} {1, 2, 1}         {0, 1,    |
|      -----------------------------------------------------------------------|
|        p          - 9750p         p          + 14311p         p          -  |
|      0} {1, 2, 1}        {0, 1, 1} {1, 2, 1}         {0, 2, 0} {1, 2, 1}    |
|      -----------------------------------------------------------------------|
|      579p         p         , p         p          - 8647p         p        |
|          {0, 2, 1} {1, 2, 1}   {0, 1, 0} {0, 2, 0}        {0, 1, 1} {0, 2,  |
|      -----------------------------------------------------------------------|
|         - 8198p         p          + 15157p         p          + 14818p     |
|      0}        {0, 1, 0} {0, 2, 1}         {0, 1, 1} {0, 2, 1}         {0,  |
|      -----------------------------------------------------------------------|
|           p          + 6619p         p          + 9296p         p          +|
|      2, 0} {1, 1, 0}        {0, 2, 1} {1, 1, 0}        {0, 2, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|      1097p         p          + 3959p         p          - 4689p            |
|           {0, 2, 1} {1, 1, 1}        {0, 1, 0} {1, 2, 0}        {0, 1,      |
|      -----------------------------------------------------------------------|
|        p          - 13227p         p          + 13254p         p          + |
|      1} {1, 2, 0}         {1, 1, 0} {1, 2, 0}         {1, 1, 1} {1, 2, 0}   |
|      -----------------------------------------------------------------------|
|      12234p         p          + 3586p         p          - 4952p           |
|            {0, 1, 0} {1, 2, 1}        {0, 1, 1} {1, 2, 1}        {1, 1,     |
|      -----------------------------------------------------------------------|
|        p          - 10474p         p         , p         p          -       |
|      0} {1, 2, 1}         {1, 1, 1} {1, 2, 1}   {0, 0, 0} {0, 2, 0}         |
|      -----------------------------------------------------------------------|
|      1636p         p          - 5775p         p          - 7412p            |
|           {0, 0, 1} {0, 2, 0}        {0, 0, 0} {0, 2, 1}        {0, 0,      |
|      -----------------------------------------------------------------------|
|        p          - 10584p         p          + 15643p         p          - |
|      1} {0, 2, 1}         {0, 2, 0} {1, 0, 0}         {0, 2, 1} {1, 0, 0}   |
|      -----------------------------------------------------------------------|
|      7986p         p          - 13762p         p          - 15076p          |
|           {0, 2, 0} {1, 0, 1}         {0, 2, 1} {1, 0, 1}         {0, 0,    |
|      -----------------------------------------------------------------------|
|        p          + 15290p         p          + 6342p         p          +  |
|      0} {1, 2, 0}         {0, 0, 1} {1, 2, 0}        {1, 0, 0} {1, 2, 0}    |
|      -----------------------------------------------------------------------|
|      8940p         p          + 9622p         p          + 7985p            |
|           {1, 0, 1} {1, 2, 0}        {0, 0, 0} {1, 2, 1}        {0, 0,      |
|      -----------------------------------------------------------------------|
|        p          - 963p         p          + 1635p         p         , p   |
|      1} {1, 2, 1}       {1, 0, 0} {1, 2, 1}        {1, 0, 1} {1, 2, 1}   {0,|
|      -----------------------------------------------------------------------|
|           p          - 13265p         p          + 342p         p          -|
|      0, 0} {0, 1, 0}         {0, 0, 1} {0, 1, 0}       {0, 0, 0} {0, 1, 1}  |
|      -----------------------------------------------------------------------|
|      12924p         p          - 2242p         p          - 1901p           |
|            {0, 0, 1} {0, 1, 1}        {0, 1, 0} {1, 0, 0}        {0, 1,     |
|      -----------------------------------------------------------------------|
|        p          + 4951p         p          + 5292p         p          -   |
|      1} {1, 0, 0}        {0, 1, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}     |
|      -----------------------------------------------------------------------|
|      6804p         p          + 11933p         p          - 9047p           |
|           {0, 0, 0} {1, 1, 0}         {0, 0, 1} {1, 1, 0}        {1, 0,     |
|      -----------------------------------------------------------------------|
|        p          - 1854p         p          - 8592p         p          +   |
|      0} {1, 1, 0}        {1, 0, 1} {1, 1, 0}        {0, 0, 0} {1, 1, 1}     |
|      -----------------------------------------------------------------------|
|      10145p         p          - 10835p         p          - 3642p          |
|            {0, 0, 1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}        {1, 0,    |
|      -----------------------------------------------------------------------|
|        p         , p         p          - 13133p         p          +       |
|      1} {1, 1, 1}   {0, 0, 0} {0, 0, 1}         {0, 0, 1} {0, 1, 0}         |
|      -----------------------------------------------------------------------|
|      2564p         p          - 10570p         p          - 12449p          |
|           {0, 0, 0} {0, 1, 1}         {0, 1, 0} {0, 1, 1}         {0, 0,    |
|      -----------------------------------------------------------------------|
|        p          - 9886p         p          - 7579p         p          +   |
|      1} {0, 2, 0}        {0, 1, 1} {0, 2, 0}        {0, 0, 0} {0, 2, 1}     |
|      -----------------------------------------------------------------------|
|      11290p         p          + 11974p         p          + 1337p          |
|            {0, 1, 0} {0, 2, 1}         {0, 2, 0} {0, 2, 1}        {0, 0,    |
|      -----------------------------------------------------------------------|
|        p          + 3900p         p          - 6243p         p          -   |
|      1} {1, 0, 0}        {0, 1, 1} {1, 0, 0}        {0, 2, 1} {1, 0, 0}     |
|      -----------------------------------------------------------------------|
|      1739p         p          - 14873p         p          - 14189p          |
|           {0, 0, 0} {1, 0, 1}         {0, 1, 0} {1, 0, 1}         {0, 2,    |
|      -----------------------------------------------------------------------|
|        p          - 403p         p          + 4755p         p          +    |
|      0} {1, 0, 1}       {1, 0, 0} {1, 0, 1}        {0, 0, 1} {1, 1, 0}      |
|      -----------------------------------------------------------------------|
|      7318p         p          - 2825p         p          + 3015p            |
|           {0, 1, 1} {1, 1, 0}        {0, 2, 1} {1, 1, 0}        {1, 0,      |
|      -----------------------------------------------------------------------|
|        p          - 3204p         p          + 15665p         p          -  |
|      1} {1, 1, 0}        {0, 0, 0} {1, 1, 1}         {0, 1, 0} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|      15654p         p          - 1868p         p          + 1550p           |
|            {0, 2, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {1, 1,     |
|      -----------------------------------------------------------------------|
|        p          + 5231p         p          + 7794p         p          -   |
|      0} {1, 1, 1}        {0, 0, 1} {1, 2, 0}        {0, 1, 1} {1, 2, 0}     |
|      -----------------------------------------------------------------------|
|      2349p         p          + 3491p         p          + 2026p            |
|           {0, 2, 1} {1, 2, 0}        {1, 0, 1} {1, 2, 0}        {1, 1,      |
|      -----------------------------------------------------------------------|
|        p          + 4832p         p          - 8302p         p          -   |
|      1} {1, 2, 0}        {0, 0, 0} {1, 2, 1}        {0, 1, 0} {1, 2, 1}     |
|      -----------------------------------------------------------------------|
|      7618p         p          + 6168p         p          + 9586p            |
|           {0, 2, 0} {1, 2, 1}        {1, 0, 0} {1, 2, 1}        {1, 1,      |
|      -----------------------------------------------------------------------|
|        p          + 10062p         p         )                              |
|      0} {1, 2, 1}         {1, 2, 0} {1, 2, 1}                               |
|                                                                             |
|o16 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+

Here is an example where the conditional independence relations are given with
a List.

+-----------------------------------------------------------------------------+
|i17 : FF = ZZ/32003                                                          |
|                                                                             |
|o17 = FF                                                                     |
|                                                                             |
|o17 : QuotientRing                                                           |
+-----------------------------------------------------------------------------+
|i18 : d = {2,2,2};                                                           |
+-----------------------------------------------------------------------------+
|i19 : X = randomGame(d, CoefficientRing => FF);                              |
+-----------------------------------------------------------------------------+
|i20 : PR = probabilityRing(d, CoefficientRing => FF);                        |
+-----------------------------------------------------------------------------+
|i21 : G = graph ({{1,2}},Singletons => {3});                                 |
+-----------------------------------------------------------------------------+
|i22 : L = {{{1,2},{3},{}}};                                                  |
+-----------------------------------------------------------------------------+
|i23 : I1 = spohnCI(PR,X,G)                                                   |
|                                                                             |
|                                                                             |
|o23 = ideal (p          - 13236p          + 10364p          - 14521p         |
|              {0, 0, 1}         {0, 1, 1}         {1, 0, 1}         {1, 1,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        , p          - 13236p          + 10364p          - 14521p         ,  |
|      1}   {0, 0, 0}         {0, 1, 0}         {1, 0, 0}         {1, 1, 0}   |
|      -----------------------------------------------------------------------|
|                                                                             |
|      p         p          - p         p         , p         p          -    |
|       {1, 0, 1} {1, 1, 0}    {1, 0, 0} {1, 1, 1}   {0, 1, 1} {1, 1, 0}      |
|      -----------------------------------------------------------------------|
|                                                                             |
|      p         p         , p         p          + 9786p         p          +|
|       {0, 1, 0} {1, 1, 1}   {0, 1, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}  |
|      -----------------------------------------------------------------------|
|                                      2                                      |
|      4771p         p          - 1915p          - 12791p         p          -|
|           {1, 0, 0} {1, 0, 1}        {1, 0, 1}         {0, 1, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      9544p         p          + 7894p         p          + 4251p            |
|           {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {1, 0,      |
|      -----------------------------------------------------------------------|
|                                                     2                       |
|        p          - 5017p         p          - 5634p         , p            |
|      1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}        {1, 1, 1}   {0, 1,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 9786p         p          + 4771p         p          -   |
|      1} {1, 0, 0}        {0, 1, 1} {1, 0, 1}        {1, 0, 0} {1, 0, 1}     |
|      -----------------------------------------------------------------------|
|           2                                                                 |
|      1915p          - 12791p         p          - 9544p         p          +|
|           {1, 0, 1}         {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      7894p         p          + 4251p         p          - 5017p            |
|           {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}        {1, 1,      |
|      -----------------------------------------------------------------------|
|                          2                                      2           |
|        p          - 5634p         , p         p          + 4771p          - |
|      0} {1, 1, 1}        {1, 1, 1}   {0, 1, 0} {1, 0, 0}        {1, 0, 0}   |
|      -----------------------------------------------------------------------|
|                                                                   2         |
|      12820p         p          + 1456p         p          - 13568p          |
|            {0, 1, 1} {1, 0, 1}        {1, 0, 0} {1, 0, 1}         {1, 0, 1} |
|      -----------------------------------------------------------------------|
|                                                                    2        |
|      - 12791p         p          + 7894p         p          - 5017p         |
|              {0, 1, 0} {1, 1, 0}        {1, 0, 0} {1, 1, 0}        {1, 1, 0}|
|      -----------------------------------------------------------------------|
|                                                                             |
|      - 551p         p          + 12830p         p          + 8809p          |
|            {0, 1, 0} {1, 1, 1}         {0, 1, 1} {1, 1, 1}        {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 3614p         p          - 1874p         p          -   |
|      0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}     |
|      -----------------------------------------------------------------------|
|           2                                       2                         |
|      6845p         , p         p          + 15038p          + 10949p        |
|           {1, 1, 1}   {0, 1, 0} {0, 1, 1}         {0, 1, 1}         {0, 1,  |
|      -----------------------------------------------------------------------|
|                                                     2                       |
|        p          - 4916p         p          - 6264p          + 14540p      |
|      1} {1, 0, 1}        {1, 0, 0} {1, 0, 1}        {1, 0, 1}         {0, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 8068p         p          - 10161p         p          -  |
|      0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                                                                   2         |
|      5702p         p          + 13601p         p          + 15402p         ,|
|           {1, 0, 1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}         {1, 1, 1} |
|      -----------------------------------------------------------------------|
|       2                2                2                                   |
|      p          - 8246p          - 4916p          + 3503p         p         |
|       {0, 1, 0}        {0, 1, 1}        {1, 0, 0}        {0, 1, 1} {1, 0, 1}|
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      - 15169p         p          - 13430p          + 14540p         p       |
|              {1, 0, 0} {1, 0, 1}         {1, 0, 1}         {0, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                             2                               |
|         - 10161p         p          + 13601p          - 12561p         p    |
|      0}         {1, 0, 0} {1, 1, 0}         {1, 1, 0}         {0, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|            + 10672p         p          - 10212p         p          -        |
|      1, 1}         {0, 1, 1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|                                                                 2           |
|      1198p         p          - 3281p         p          + 7320p         )  |
|           {1, 0, 1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}        {1, 1, 1}   |
|                                                                             |
|o23 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i24 : I2 = spohnCI(PR,X,L)                                                   |
|                                                                             |
|                                                                             |
|o24 = ideal (p          - 13236p          + 10364p          - 14521p         |
|              {0, 0, 1}         {0, 1, 1}         {1, 0, 1}         {1, 1,   |
|      -----------------------------------------------------------------------|
|                                                                             |
|        , p          - 13236p          + 10364p          - 14521p         ,  |
|      1}   {0, 0, 0}         {0, 1, 0}         {1, 0, 0}         {1, 1, 0}   |
|      -----------------------------------------------------------------------|
|                                                                             |
|      p         p          - p         p         , p         p          -    |
|       {1, 0, 1} {1, 1, 0}    {1, 0, 0} {1, 1, 1}   {0, 1, 1} {1, 1, 0}      |
|      -----------------------------------------------------------------------|
|                                                                             |
|      p         p         , p         p          + 9786p         p          +|
|       {0, 1, 0} {1, 1, 1}   {0, 1, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}  |
|      -----------------------------------------------------------------------|
|                                      2                                      |
|      4771p         p          - 1915p          - 12791p         p          -|
|           {1, 0, 0} {1, 0, 1}        {1, 0, 1}         {0, 1, 0} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      9544p         p          + 7894p         p          + 4251p            |
|           {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {1, 0,      |
|      -----------------------------------------------------------------------|
|                                                     2                       |
|        p          - 5017p         p          - 5634p         , p            |
|      1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}        {1, 1, 1}   {0, 1,      |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 9786p         p          + 4771p         p          -   |
|      1} {1, 0, 0}        {0, 1, 1} {1, 0, 1}        {1, 0, 0} {1, 0, 1}     |
|      -----------------------------------------------------------------------|
|           2                                                                 |
|      1915p          - 12791p         p          - 9544p         p          +|
|           {1, 0, 1}         {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}  |
|      -----------------------------------------------------------------------|
|                                                                             |
|      7894p         p          + 4251p         p          - 5017p            |
|           {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}        {1, 1,      |
|      -----------------------------------------------------------------------|
|                          2                                      2           |
|        p          - 5634p         , p         p          + 4771p          - |
|      0} {1, 1, 1}        {1, 1, 1}   {0, 1, 0} {1, 0, 0}        {1, 0, 0}   |
|      -----------------------------------------------------------------------|
|                                                                   2         |
|      12820p         p          + 1456p         p          - 13568p          |
|            {0, 1, 1} {1, 0, 1}        {1, 0, 0} {1, 0, 1}         {1, 0, 1} |
|      -----------------------------------------------------------------------|
|                                                                    2        |
|      - 12791p         p          + 7894p         p          - 5017p         |
|              {0, 1, 0} {1, 1, 0}        {1, 0, 0} {1, 1, 0}        {1, 1, 0}|
|      -----------------------------------------------------------------------|
|                                                                             |
|      - 551p         p          + 12830p         p          + 8809p          |
|            {0, 1, 0} {1, 1, 1}         {0, 1, 1} {1, 1, 1}        {1, 0,    |
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          + 3614p         p          - 1874p         p          -   |
|      0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}     |
|      -----------------------------------------------------------------------|
|           2                                       2                         |
|      6845p         , p         p          + 15038p          + 10949p        |
|           {1, 1, 1}   {0, 1, 0} {0, 1, 1}         {0, 1, 1}         {0, 1,  |
|      -----------------------------------------------------------------------|
|                                                     2                       |
|        p          - 4916p         p          - 6264p          + 14540p      |
|      1} {1, 0, 1}        {1, 0, 0} {1, 0, 1}        {1, 0, 1}         {0, 1,|
|      -----------------------------------------------------------------------|
|                                                                             |
|        p          - 8068p         p          - 10161p         p          -  |
|      0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}    |
|      -----------------------------------------------------------------------|
|                                                                   2         |
|      5702p         p          + 13601p         p          + 15402p         ,|
|           {1, 0, 1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}         {1, 1, 1} |
|      -----------------------------------------------------------------------|
|       2                2                2                                   |
|      p          - 8246p          - 4916p          + 3503p         p         |
|       {0, 1, 0}        {0, 1, 1}        {1, 0, 0}        {0, 1, 1} {1, 0, 1}|
|      -----------------------------------------------------------------------|
|                                          2                                  |
|      - 15169p         p          - 13430p          + 14540p         p       |
|              {1, 0, 0} {1, 0, 1}         {1, 0, 1}         {0, 1, 0} {1, 1, |
|      -----------------------------------------------------------------------|
|                                             2                               |
|         - 10161p         p          + 13601p          - 12561p         p    |
|      0}         {1, 0, 0} {1, 1, 0}         {1, 1, 0}         {0, 1, 0} {1, |
|      -----------------------------------------------------------------------|
|                                                                             |
|            + 10672p         p          - 10212p         p          -        |
|      1, 1}         {0, 1, 1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}          |
|      -----------------------------------------------------------------------|
|                                                                 2           |
|      1198p         p          - 3281p         p          + 7320p         )  |
|           {1, 0, 1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}        {1, 1, 1}   |
|                                                                             |
|o24 : Ideal of PR                                                            |
+-----------------------------------------------------------------------------+
|i25 : I1 == I2                                                               |
|                                                                             |
|o25 = true                                                                   |
+-----------------------------------------------------------------------------+

The Verbose=>true option prints the progress of each step in the saturation
process - a message is printed after saturating the ideal of the Spohn variety
$V$, the conditional independence ideal $I$, and the sum $V + I$ with respect
to each hyperplane of the probability simplex.

+-----------------------------------------------------+
|i26 : FF = ZZ/32003;                                 |
+-----------------------------------------------------+
|i27 : d = {2,3,2};                                   |
+-----------------------------------------------------+
|i28 : X = randomGame(d, CoefficientRing => FF);      |
+-----------------------------------------------------+
|i29 : PR = probabilityRing(d, CoefficientRing => FF);|
+-----------------------------------------------------+
|i30 : L = {{{1,2},{3},{}}};                          |
+-----------------------------------------------------+
|i31 : I = spohnCI(PR, X, L, Verbose=>true);          |
|Completed step 1 of saturating CI ideal              |
|Completed step 1 of saturating input ideal           |
|Completed step 2 of saturating CI ideal              |
|Completed step 2 of saturating input ideal           |
|Completed step 3 of saturating CI ideal              |
|Completed step 3 of saturating input ideal           |
|Completed step 4 of saturating CI ideal              |
|Completed step 4 of saturating input ideal           |
|Completed step 5 of saturating CI ideal              |
|Completed step 5 of saturating input ideal           |
|Completed step 6 of saturating CI ideal              |
|Completed step 6 of saturating input ideal           |
|Completed step 7 of saturating CI ideal              |
|Completed step 7 of saturating input ideal           |
|Completed step 8 of saturating CI ideal              |
|Completed step 8 of saturating input ideal           |
|Completed step 9 of saturating CI ideal              |
|Completed step 9 of saturating input ideal           |
|Completed step 10 of saturating CI ideal             |
|Completed step 10 of saturating input ideal          |
|Completed step 11 of saturating CI ideal             |
|Completed step 11 of saturating input ideal          |
|Completed step 12 of saturating CI ideal             |
|Completed step 12 of saturating input ideal          |
|Completed step 13 of saturating CI ideal             |
|Completed step 13 of saturating input ideal          |
|Completed step 1 of saturating sum                   |
|Completed step 2 of saturating sum                   |
|Completed step 3 of saturating sum                   |
|Completed step 4 of saturating sum                   |
|Completed step 5 of saturating sum                   |
|Completed step 6 of saturating sum                   |
|Completed step 7 of saturating sum                   |
|Completed step 8 of saturating sum                   |
|Completed step 9 of saturating sum                   |
|Completed step 10 of saturating sum                  |
|Completed step 11 of saturating sum                  |
|Completed step 12 of saturating sum                  |
|                                                     |
|o31 : Ideal of PR                                    |
+-----------------------------------------------------+

See also
========

  * spohnIdeal -- compute the ideal of the Spohn variety of
    a given game
  * ciIdeal -- the ideal of a list of conditional independence
    statements
  * intersectWithCImodel -- ideal of the
    intersection of a given variety with the conditional independence model
  * conditionalIndependenceIdeal (GraphicalModels)
    -- ideal of a list of
    conditional independent statements

Ways to use spohnCI:
====================

  * "spohnCI(Ring,List,Graph)"
  * "spohnCI(Ring,List,Graph,List)"
  * "spohnCI(Ring,List,List)"
  * "spohnCI(Ring,List,List,List)"

For the programmer
==================

The object spohnCI, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:2169:0.

spohnIdeal -- compute the ideal of the Spohn variety of a given game
********************************************************************

  * Usage:
        spohnIdeal(PR,X)
  * Inputs:
      * PR, a ring (Macaulay2Doc), a probability ring obtained via
        probabilityRing(Di), where $Di$ is the format of the game
      * X, a list (Macaulay2Doc), a list of n tensors of format
        $D_i$ describing the payoffs of the game
  * Outputs:
      * a list (Macaulay2Doc), the ideal generated by the $2\times
        2$ minors of the Spohn matrices of the game $X$

Description
===========

It is crucial that the formats in PR and X match up. The Spohn ideal $I_X$ is
the ideal defining the Spohn variety of a game $X$, which contains the
dependency equilibria of the game $X$. Its generators are given by the $2\times
2$ minors of the Spohn matrices. This function uses the function spohnMatrices
to compute the Spohn matrices of the given game.

+-----------------------------------------------------------------------------+
|i1 : Di = {2,2,3};                                                           |
+-----------------------------------------------------------------------------+
|i2 : PR = probabilityRing(Di);                                               |
+-----------------------------------------------------------------------------+
|i3 : X = randomGame(Di);                                                     |
+-----------------------------------------------------------------------------+
|i4 : I = spohnIdeal(PR,X)                                                    |
|                                                                             |
|              19                       31                        1           |
|o4 = ideal (- --p         p          - --p         p          - --p          |
|               5 {0, 0, 0} {1, 0, 0}   20 {0, 0, 1} {1, 0, 0}   20 {0, 0,    |
|     ------------------------------------------------------------------------|
|                    21                        7                       13     |
|       p          - --p         p          - --p         p          - --p    |
|     2} {1, 0, 0}   20 {0, 1, 0} {1, 0, 0}   90 {0, 1, 1} {1, 0, 0}    6 {0, |
|     ------------------------------------------------------------------------|
|                        1                       19                           |
|          p          + --p         p          + --p         p          +     |
|     0, 0} {1, 0, 1}   12 {0, 0, 1} {1, 0, 1}   12 {0, 0, 2} {1, 0, 1}       |
|     ------------------------------------------------------------------------|
|      7                       14                       49                    |
|     --p         p          + --p         p          + --p         p         |
|     12 {0, 1, 0} {1, 0, 1}    9 {0, 1, 1} {1, 0, 1}   30 {0, 1, 2} {1, 0, 1}|
|     ------------------------------------------------------------------------|
|       5                       19                       25                   |
|     + -p         p          + --p         p          + --p         p        |
|       2 {0, 0, 0} {1, 0, 2}    4 {0, 0, 1} {1, 0, 2}    4 {0, 0, 2} {1, 0,  |
|     ------------------------------------------------------------------------|
|          21                       56                       63               |
|        + --p         p          + --p         p          + --p         p    |
|     2}    4 {0, 1, 0} {1, 0, 2}    9 {0, 1, 1} {1, 0, 2}   10 {0, 1, 2} {1, |
|     ------------------------------------------------------------------------|
|             57                       51                        9            |
|           - --p         p          - --p         p          - --p           |
|     0, 2}   14 {0, 0, 0} {1, 1, 0}   28 {0, 0, 1} {1, 1, 0}   28 {0, 0,     |
|     ------------------------------------------------------------------------|
|                    37                       22                       19     |
|       p          - --p         p          - --p         p          - --p    |
|     2} {1, 1, 0}   28 {0, 1, 0} {1, 1, 0}   63 {0, 1, 1} {1, 1, 0}   70 {0, |
|     ------------------------------------------------------------------------|
|                       51                       39                           |
|          p          - --p         p          - --p         p          +     |
|     1, 2} {1, 1, 0}   14 {0, 0, 0} {1, 1, 1}   28 {0, 0, 1} {1, 1, 1}       |
|     ------------------------------------------------------------------------|
|      3                       25                        5                    |
|     --p         p          - --p         p          + --p         p         |
|     28 {0, 0, 2} {1, 1, 1}   28 {0, 1, 0} {1, 1, 1}   63 {0, 1, 1} {1, 1, 1}|
|     ------------------------------------------------------------------------|
|       11                       3                       15                   |
|     + --p         p          + -p         p          + --p         p        |
|       70 {0, 1, 2} {1, 1, 1}   2 {0, 0, 0} {1, 1, 2}    4 {0, 0, 1} {1, 1,  |
|     ------------------------------------------------------------------------|
|          21                       17                       47               |
|        + --p         p          + --p         p          + --p         p    |
|     2}    4 {0, 0, 2} {1, 1, 2}    4 {0, 1, 0} {1, 1, 2}    9 {0, 1, 1} {1, |
|     ------------------------------------------------------------------------|
|             53                        23                       13           |
|           + --p         p         , - --p         p          + --p          |
|     1, 2}   10 {0, 1, 2} {1, 1, 2}    28 {0, 0, 0} {0, 1, 0}   63 {0, 0,    |
|     ------------------------------------------------------------------------|
|                     9                       15                       43     |
|       p          + --p         p          + --p         p          + --p    |
|     1} {0, 1, 0}   70 {0, 0, 2} {0, 1, 0}    4 {0, 0, 0} {0, 1, 1}    9 {0, |
|     ------------------------------------------------------------------------|
|                       47                        5                           |
|          p          + --p         p          - --p         p          +     |
|     0, 1} {0, 1, 1}   10 {0, 0, 2} {0, 1, 1}   36 {0, 0, 0} {0, 1, 2}       |
|     ------------------------------------------------------------------------|
|     8                       73                       67                     |
|     -p         p          + --p         p          - --p         p          |
|     9 {0, 0, 1} {0, 1, 2}   90 {0, 0, 2} {0, 1, 2}    7 {0, 1, 0} {1, 0, 0} |
|     ------------------------------------------------------------------------|
|                               80                       15                   |
|     - 5p         p          - --p         p          - --p         p        |
|         {0, 1, 1} {1, 0, 0}    9 {0, 1, 2} {1, 0, 0}   14 {0, 1, 0} {1, 0,  |
|     ------------------------------------------------------------------------|
|          7                        7                       25                |
|        + -p         p          - --p         p          - --p         p     |
|     1}   2 {0, 1, 1} {1, 0, 1}   18 {0, 1, 2} {1, 0, 1}   56 {0, 1, 0} {1,  |
|     ------------------------------------------------------------------------|
|             33                       17                        5            |
|           + --p         p          + --p         p          - --p           |
|     0, 2}    8 {0, 1, 1} {1, 0, 2}   72 {0, 1, 2} {1, 0, 2}   12 {0, 0,     |
|     ------------------------------------------------------------------------|
|                    11                        8                       55     |
|       p          + --p         p          + --p         p          - --p    |
|     0} {1, 1, 0}   18 {0, 0, 1} {1, 1, 0}   15 {0, 0, 2} {1, 1, 0}    6 {1, |
|     ------------------------------------------------------------------------|
|                       2                        1                            |
|          p          - -p         p          - --p         p          +      |
|     0, 0} {1, 1, 0}   3 {1, 0, 1} {1, 1, 0}   24 {1, 0, 2} {1, 1, 0}        |
|     ------------------------------------------------------------------------|
|     15                       43                       47                    |
|     --p         p          + --p         p          + --p         p         |
|      4 {0, 0, 0} {1, 1, 1}    9 {0, 0, 1} {1, 1, 1}   10 {0, 0, 2} {1, 1, 1}|
|     ------------------------------------------------------------------------|
|                               7                       33                    |
|     - 5p         p          + -p         p          + --p         p         |
|         {1, 0, 0} {1, 1, 1}   2 {1, 0, 1} {1, 1, 1}    8 {1, 0, 2} {1, 1, 1}|
|     ------------------------------------------------------------------------|
|       17                        8                        1                  |
|     - --p         p          + --p         p          + --p         p       |
|       20 {0, 0, 0} {1, 1, 2}   45 {0, 0, 1} {1, 1, 2}   10 {0, 0, 2} {1, 1, |
|     ------------------------------------------------------------------------|
|          48                       11                       19               |
|        - --p         p          - --p         p          - --p         p    |
|     2}    5 {1, 0, 0} {1, 1, 2}   10 {1, 0, 1} {1, 1, 2}   40 {1, 0, 2} {1, |
|     ------------------------------------------------------------------------|
|            11                       23                       20             |
|          , --p         p          + --p         p          - --p            |
|     1, 2}   6 {0, 0, 0} {0, 0, 1}   10 {0, 0, 1} {0, 1, 0}   21 {0, 0,      |
|     ------------------------------------------------------------------------|
|                    17                       11                       20     |
|       p          - --p         p          + --p         p          - --p    |
|     0} {0, 1, 1}   35 {0, 1, 0} {0, 1, 1}    6 {0, 0, 1} {1, 0, 0}   21 {0, |
|     ------------------------------------------------------------------------|
|                       47                       11                           |
|          p          - --p         p          - --p         p          -     |
|     1, 1} {1, 0, 0}   30 {0, 0, 0} {1, 0, 1}   10 {0, 1, 0} {1, 0, 1}       |
|     ------------------------------------------------------------------------|
|     47                       43                       2                     |
|     --p         p          + --p         p          + -p         p          |
|     30 {1, 0, 0} {1, 0, 1}   14 {0, 0, 1} {1, 1, 0}   7 {0, 1, 1} {1, 1, 0} |
|     ------------------------------------------------------------------------|
|       23                       23                        3                  |
|     - --p         p          - --p         p          - --p         p       |
|       70 {1, 0, 1} {1, 1, 0}   30 {0, 0, 0} {1, 1, 1}   10 {0, 1, 0} {1, 1, |
|     ------------------------------------------------------------------------|
|          23                       33                        19              |
|        - --p         p          + --p         p         , - --p         p   |
|     1}   30 {1, 0, 0} {1, 1, 1}   70 {1, 1, 0} {1, 1, 1}    15 {0, 0, 0} {0,|
|     ------------------------------------------------------------------------|
|             4                       10                       29             |
|           - -p         p          - --p         p          - --p            |
|     0, 2}   5 {0, 0, 2} {0, 1, 0}    9 {0, 0, 0} {0, 1, 2}   45 {0, 1,      |
|     ------------------------------------------------------------------------|
|                    19                       10                       1      |
|       p          - --p         p          - --p         p          - -p     |
|     0} {0, 1, 2}   15 {0, 0, 2} {1, 0, 0}    9 {0, 1, 2} {1, 0, 0}   3 {0,  |
|     ------------------------------------------------------------------------|
|                        2                       1                            |
|          p          + --p         p          - -p         p          -      |
|     0, 0} {1, 0, 2}   15 {0, 1, 0} {1, 0, 2}   3 {1, 0, 0} {1, 0, 2}        |
|     ------------------------------------------------------------------------|
|      1                        8                       19                    |
|     --p         p          + --p         p          + --p         p         |
|     35 {0, 0, 2} {1, 1, 0}   63 {0, 1, 2} {1, 1, 0}   21 {1, 0, 2} {1, 1, 0}|
|     ------------------------------------------------------------------------|
|       23                       22                       23                  |
|     - --p         p          - --p         p          - --p         p       |
|       21 {0, 0, 0} {1, 1, 2}   35 {0, 1, 0} {1, 1, 2}   21 {1, 0, 0} {1, 1, |
|     ------------------------------------------------------------------------|
|          1                        31                       11               |
|        + -p         p         , - --p         p          - --p         p    |
|     2}   7 {1, 1, 0} {1, 1, 2}    10 {0, 0, 1} {0, 0, 2}   35 {0, 0, 2} {0, |
|     ------------------------------------------------------------------------|
|             53                       10                        3            |
|           - --p         p          - --p         p          + --p           |
|     1, 1}   18 {0, 0, 1} {0, 1, 2}   63 {0, 1, 1} {0, 1, 2}   10 {0, 0,     |
|     ------------------------------------------------------------------------|
|                    41                       13                       13     |
|       p          + --p         p          - --p         p          + --p    |
|     2} {1, 0, 1}   90 {0, 1, 2} {1, 0, 1}    6 {0, 0, 1} {1, 0, 2}   21 {0, |
|     ------------------------------------------------------------------------|
|                       37                       1                            |
|          p          + --p         p          - -p         p          -      |
|     1, 1} {1, 0, 2}   30 {1, 0, 1} {1, 0, 2}   2 {0, 0, 2} {1, 1, 1}        |
|     ------------------------------------------------------------------------|
|     31                       13                       41                    |
|     --p         p          + --p         p          - --p         p         |
|     90 {0, 1, 2} {1, 1, 1}   30 {1, 0, 2} {1, 1, 1}   14 {0, 0, 1} {1, 1, 2}|
|     ------------------------------------------------------------------------|
|       1                       33                       23                   |
|     - -p         p          + --p         p          - --p         p        |
|       7 {0, 1, 1} {1, 1, 2}   70 {1, 0, 1} {1, 1, 2}   70 {1, 1, 1} {1, 1,  |
|     ------------------------------------------------------------------------|
|       )                                                                     |
|     2}                                                                      |
|                                                                             |
|o4 : Ideal of PR                                                             |
+-----------------------------------------------------------------------------+

See also
========

  * probabilityRing -- ring of probability
    distributions of a game indexed by ordered multi-indices
  * randomGame -- construct a game of a given format with
    arbitrary payoffs
  * spohnMatrices -- compute the list of Spohn matrices
    of a given game
  * konstanzMatrix -- construct the Konstanz matrix of a
    given game

Ways to use spohnIdeal:
=======================

  * spohnIdeal(Ring,List) (missing documentation)

For the programmer
==================

The object spohnIdeal, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1695:0.

spohnMatrices -- compute the list of Spohn matrices of a given game
*******************************************************************

  * Usage:
        spohnMatrices(PR,X)
  * Inputs:
      * PR, a ring (Macaulay2Doc), a probability ring obtained via
        probabilityRing(Di), where $Di$ is the format of the game
      * X, a list (Macaulay2Doc), a list of n tensors of format
        $D_i$ describing the payoffs of the game
  * Outputs:
      * a list (Macaulay2Doc), the list of n Spohn matrices
        describing the dependency equilibria of the game $X$

Description
===========

It is crucial that the formats in PR and X match up. The Spohn matrix $M_i$ is
the $d_i \times 2$ matrix encoding the denominators and nominators of the
conditional expected payoffs of the $i$-th player. The Spohn matrices have rank
one at the dependency equilibria of the game $X$.

+-----------------------------------------------------------------------------+
|i1 : Di = {2,2,3};                                                           |
+-----------------------------------------------------------------------------+
|i2 : PR = probabilityRing(Di);                                               |
+-----------------------------------------------------------------------------+
|i3 : X = randomGame(Di);                                                     |
+-----------------------------------------------------------------------------+
|i4 : M = spohnMatrices(PR,X)                                                 |
|                                                                             |
|o4 = {| p_{0, 0, 0}+p_{0, 0, 1}+p_{0, 0, 2}+p_{0, 1, 0}+p_{0, 1, 1}+p_{0, 1, |
|      | p_{1, 0, 0}+p_{1, 0, 1}+p_{1, 0, 2}+p_{1, 1, 0}+p_{1, 1, 1}+p_{1, 1, |
|                                                                             |
|     ------------------------------------------------------------------------|
|     2} 9/2p_{0, 0, 0}+9/4p_{0, 0, 1}+3/4p_{0, 0, 2}+7/4p_{0, 1, 0}+7/9p_{0, |
|     2} 7/10p_{1, 0, 0}+7/3p_{1, 0, 1}+7p_{1, 0, 2}+3/7p_{1, 1, 0}+6/7p_{1, 1|
|                                                                             |
|     ------------------------------------------------------------------------|
|     1, 1}+7/10p_{0, 1, 2} |, | p_{0, 0, 0}+p_{0, 0, 1}+p_{0, 0, 2}+p_{1, 0, |
|     , 1}+6p_{1, 1, 2}     |  | p_{0, 1, 0}+p_{0, 1, 1}+p_{0, 1, 2}+p_{1, 1, |
|                                                                             |
|     ------------------------------------------------------------------------|
|     0}+p_{1, 0, 1}+p_{1, 0, 2} 5/4p_{0, 0,                                  |
|     0}+p_{1, 1, 1}+p_{1, 1, 2} 3/7p_{0, 1,                                  |
|                                                                             |
|     ------------------------------------------------------------------------|
|     0}+2/9p_{0, 0, 1}+3/10p_{0, 0, 2}+10p_{1, 0, 0}+3/2p_{1, 0, 1}+7/8p_{1, |
|     0}+5p_{0, 1, 1}+10/9p_{0, 1, 2}+5/6p_{1, 1, 0}+5p_{1, 1, 1}+2/5p_{1, 1, |
|                                                                             |
|     ------------------------------------------------------------------------|
|     0, 2} |, | p_{0, 0, 0}+p_{0, 1, 0}+p_{1, 0, 0}+p_{1, 1, 0} 5/3p_{0, 0,  |
|     2}    |  | p_{0, 0, 1}+p_{0, 1, 1}+p_{1, 0, 1}+p_{1, 1, 1} 7/2p_{0, 0,  |
|              | p_{0, 0, 2}+p_{0, 1, 2}+p_{1, 0, 2}+p_{1, 1, 2} 2/5p_{0, 0,  |
|     ------------------------------------------------------------------------|
|     0}+6/5p_{0, 1, 0}+5/3p_{1, 0, 0}+3/7p_{1, 1, 0}   |}                    |
|     1}+5/7p_{0, 1, 1}+1/10p_{1, 0, 1}+9/10p_{1, 1, 1} |                     |
|     2}+5/9p_{0, 1, 2}+4/3p_{1, 0, 2}+4/7p_{1, 1, 2}   |                     |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * probabilityRing -- ring of probability
    distributions of a game indexed by ordered multi-indices
  * randomGame -- construct a game of a given format with
    arbitrary payoffs
  * spohnIdeal -- compute the ideal of the Spohn variety of
    a given game
  * konstanzMatrix -- construct the Konstanz matrix of a
    given game

Ways to use spohnMatrices:
==========================

  * spohnMatrices(Ring,List) (missing documentation)

For the programmer
==================

The object spohnMatrices, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1658:0.

Tensor -- a mutable hash table representing a tensor
****************************************************

  * Usage:
        T = zeroTensor(R, d)
  * Outputs:
      * an instance of the type Tensor, A Tensor object storing
        values at multi-indices.

Description
===========

The type `Tensor` is a mutable hash table with additional metadata to represent
multi-dimensional arrays. Each tensor has an associated format (list of
dimensions), a coefficient ring, and a set of index keys.

+-------------------------------------+
|i1 : T = zeroTensor(QQ, {2,2});      |
+-------------------------------------+
|i2 : T#{0,0} = 1;                    |
+-------------------------------------+
|i3 : T#{1,1} = 5;                    |
+-------------------------------------+
|i4 : format T                        |
|                                     |
|o4 = {2, 2}                          |
|                                     |
|o4 : List                            |
+-------------------------------------+
|i5 : coefficientRing T               |
|                                     |
|o5 = QQ                              |
|                                     |
|o5 : Ring                            |
+-------------------------------------+
|i6 : indexset T                      |
|                                     |
|o6 = {{0, 0}, {0, 1}, {1, 0}, {1, 1}}|
|                                     |
|o6 : List                            |
+-------------------------------------+

See also
========

  * zeroTensor -- construct a tensor with zero entries from
    a given ring.
  * randomTensor -- construct a tensor with random entries
    from a given ring

Methods that use an object of class Tensor:
===========================================

  * "assemblePlayeriPolynomials(Ring,Tensor,ZZ)" -- see
    assemblePlayeriPolynomials, -- get all
    incentive constraint polynomials for a player
  * "assemblePolynomial(Ring,Tensor,List)" -- see assemblePolynomial,
    -- compute incentive constraint polynomial for
    deviation
  * coefficientRing(Tensor) (missing documentation)
  * format(Tensor) (missing documentation)
  * "indexset(Tensor)" -- see indexset, -- get the list of
    index tuples of a tensor
  * "slice(Tensor,List,List)" -- see slice, -- extract a slice of
    a tensor

For the programmer
==================

The object Tensor, is a type (Macaulay2Doc), with
ancestor classes MutableHashTable (Macaulay2Doc), <
HashTable (Macaulay2Doc) < Thing (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:986:0.

toMarkovRing -- ring of joint probability distributions created with the markovRing function from the GraphicalModels package
*****************************************************************************************************************************

  * Usage:
        toMarkovRing R
  * Inputs:
      * R, a polynomial ring (Macaulay2Doc), created
        using the probabilityRing method
  * Outputs:
      * a polynomial ring (Macaulay2Doc), a polynomial
        ring isomorphic to the input ring created by the markovRing method from
        the GraphicalModels package, with variables $q_{(i_1+1, \dots ,
        i_k+1)}$ corresponding to the variables $p_{\{i_1, \ldots, i_k\}}$ of
        the input ring

Description
===========

Given a ring created with the probabilityRing function, this function creates
the canonically isomorphic ring defined by the markovRing function from the
GraphicalModels package. The variable name of the output ring is set to be
different from the variable name of the input ring: the default variable name
of the output ring is "p", and if the variable name of the input ring is "p"
then the variable name of the output ring becomes "q".

+-----------------------------------------------------------------------------+
|i1 : R = probabilityRing({2,3,4}, CoefficientRing => ZZ/32003,               |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
|-----------------------------------------------------------------------------|
|ProbabilityVariableName => "x")                                              |
+-----------------------------------------------------------------------------+
|i2 : markovR = toMarkovRing R;                                               |
+-----------------------------------------------------------------------------+
|i3 : numgens markovR                                                         |
|                                                                             |
|o3 = 24                                                                      |
+-----------------------------------------------------------------------------+
|i4 : R_0, R_11, R_23                                                         |
|                                                                             |
|o4 = (x         , x         , x         )                                    |
|       {0, 0, 0}   {0, 2, 3}   {1, 2, 3}                                     |
|                                                                             |
|o4 : Sequence                                                                |
+-----------------------------------------------------------------------------+

See also
========

  * probabilityRing -- ring of probability
    distributions of a game indexed by ordered multi-indices
  * gaussianRing (GraphicalModels) -- ring of Gaussian
    correlations on n random variables or a graphical model

Ways to use toMarkovRing:
=========================

  * "toMarkovRing(Ring)"

For the programmer
==================

The object toMarkovRing, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1779:0.

zeroTensor -- construct a tensor with zero entries from a given ring.
*********************************************************************

  * Usage:
        zeroTensor format
        zeroTensor(R, format)
  * Inputs:
      * format, a list (Macaulay2Doc), A list of integers
        specifying the format of the tensor (e.g., {2,2,2} creates a 2x2x2
        tensor with zero entries).
      * R, a ring (Macaulay2Doc), A ring from which zeros will be
        drawn (optional).
  * Outputs:
      * an instance of the type Tensor, A tensor whose entries
        are all zero.

Description
===========

This method constructs a tensor with the specified format and fills it with
zeros from the given ring. Internally, it uses a hash table where each key is a
multi-index (a list of positions). Metadata such as the format, coefficient
ring, and index set are stored in the tensor as well. It is useful for defining
a custom tensor.

+--------------------------------------------------------+
|i1 : T = zeroTensor {2,2};                              |
+--------------------------------------------------------+
|i2 : T#{0,0} = 1;                                       |
+--------------------------------------------------------+
|i3 : T#{0,1} = 2;                                       |
+--------------------------------------------------------+
|i4 : T#{1,0} = 3;                                       |
+--------------------------------------------------------+
|i5 : T#{1,1} = 4;                                       |
+--------------------------------------------------------+
|i6 : format T                                           |
|                                                        |
|o6 = {2, 2}                                             |
|                                                        |
|o6 : List                                               |
+--------------------------------------------------------+
|i7 : peek T                                             |
|                                                        |
|o7 = Tensor{{0, 0} => 1                                }|
|            {0, 1} => 2                                 |
|            {1, 0} => 3                                 |
|            {1, 1} => 4                                 |
|            coefficients => QQ                          |
|            format => {2, 2}                            |
|            indexes => {{0, 0}, {0, 1}, {1, 0}, {1, 1}} |
+--------------------------------------------------------+

See also
========

  * randomTensor -- construct a tensor with random entries
    from a given ring

Ways to use zeroTensor:
=======================

  * "zeroTensor(List)"
  * "zeroTensor(Ring,List)"

For the programmer
==================

The object zeroTensor, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in GameTheory.m2:1068:0.