Macaulay2 » Documentation
Packages » PolyominoIdeals » equivalenceClassesSwitchingRook
next | previous | forward | backward | up | index | toc

equivalenceClassesSwitchingRook -- Equivalence classes of rook configurations under switching

Description

This function computes the equivalence classes of non-attacking rook configurations under the switching operation.

Let $\mathcal{P}$ be a collection of cells. Two rooks are non-attacking if they do not share the same row or column or, if they are on the same row (resp. column), they are not connected by a horizontal (resp. vertical) path of cells of $\mathcal{P}$ A $j$-rook configuration in $\mathcal{P}$ is a set of $j$ non-attacking rooks placed within $\mathcal{P}$, where $j \ge 0$; by convention, the $0$-rook configuration is the empty set $\emptyset$. We denote by $\mathcal{R}_j(\mathcal{P})$ the set of all $j$-rook configurations of $\mathcal{P}$ for $j \in \{0, \dots, r(\mathcal{P})\}$, adopting the convention that $\mathcal{R}_0(\mathcal{P}) = \{\emptyset\}$ (so $|\mathcal{R}_0(\mathcal{P})| = 1$).
The collection $\bigcup_{j=0}^{r(\mathcal{P})} \mathcal{R}_j(\mathcal{P})$ forms a simplicial complex, called the chessboard complex of $\mathcal{P}$.

Two non-attacking rooks in $\mathcal{P}$ are said to be in switching position if they occupy the diagonal (respectively, anti-diagonal) corner cells of an inner interval $I$ of $\mathcal{P}$. In that case, we say the rooks are in a diagonal (respectively, anti-diagonal) position.
Fix $j \in \{0, \dots, r(\mathcal{P})\}$ and let $F \in \mathcal{R}_j(\mathcal{P})$.
Suppose that two rooks $R_1, R_2 \in F$ are in switching position in $I$, for some inner interval $I$. Let $R_1', R_2'$ denote the rooks in the opposite diagonal (or anti-diagonal) cells of $I$. Then the configuration $(F \setminus \{R_1, R_2\}) \cup \{R_1', R_2'\}$ also belongs to $\mathcal{R}_j(\mathcal{P})$. This operation of replacing $R_1, R_2$ with $R_1', R_2'$ is called a switch of $R_1$ and $R_2$.
It defines an equivalence relation $\sim$ on $\mathcal{R}_j(\mathcal{P})$, where $F_1 \sim F_2$ if and only if $F_2$ can be obtained from $F_1$ by a finite sequence of switches.

i1 : Q = cellCollection {{1,1},{1,2},{2,1},{2,2}};
 
i2 : netList equivalenceClassesSwitchingRook Q

     +------------------------------------+----------+----------+----------+
o2 = |{{{1, 1}}}                          |{{{1, 2}}}|{{{2, 1}}}|{{{2, 2}}}|
     +------------------------------------+----------+----------+----------+
     |{{{1, 2}, {2, 1}}, {{1, 1}, {2, 2}}}|          |          |          |
     +------------------------------------+----------+----------+----------+
 


i3 : Q = cellCollection {{1,1},{1,2},{2,1},{2,2},{1,3},{2,3},{3,1}};
 
i4 : netList equivalenceClassesSwitchingRook Q

     +----------------------------------------------------+------------------------------------+------------------+------------------------------------+------------------------------------+------------------+------------------+
o4 = |{{{1, 1}}}                                          |{{{1, 2}}}                          |{{{1, 3}}}        |{{{2, 1}}}                          |{{{2, 2}}}                          |{{{2, 3}}}        |{{{3, 1}}}        |
     +----------------------------------------------------+------------------------------------+------------------+------------------------------------+------------------------------------+------------------+------------------+
     |{{{1, 3}, {3, 1}}}                                  |{{{1, 2}, {2, 3}}, {{1, 3}, {2, 2}}}|{{{2, 3}, {3, 1}}}|{{{1, 1}, {2, 2}}, {{1, 2}, {2, 1}}}|{{{1, 1}, {2, 3}}, {{1, 3}, {2, 1}}}|{{{1, 2}, {3, 1}}}|{{{2, 2}, {3, 1}}}|
     +----------------------------------------------------+------------------------------------+------------------+------------------------------------+------------------------------------+------------------+------------------+
     |{{{1, 2}, {2, 3}, {3, 1}}, {{1, 3}, {2, 2}, {3, 1}}}|                                    |                  |                                    |                                    |                  |                  |
     +----------------------------------------------------+------------------------------------+------------------+------------------------------------+------------------------------------+------------------+------------------+

See also

Ways to use equivalenceClassesSwitchingRook:

  • equivalenceClassesSwitchingRook(CollectionOfCells)

For the programmer

The object equivalenceClassesSwitchingRook is a method function with options.

The source of this document is in PolyominoIdeals.m2:1443:0.