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# fVector(SimplicialComplex) -- compute the f-vector of an abstract simplicial complex

## Synopsis

• Function: fVector
• Usage:
f = fVector Delta
• Inputs:
• Delta, ,
• Outputs:
• f, a list, where the $i$-th entry is the number of faces in $\Delta$ of dimension $i-1$ and $0 \leqslant i \leqslant \operatorname{dim} \Delta$

## Description

The f-vector of an abstract simplicial complex is the vector $(f_{-1}, f_0, f_1, \dotsc, f_d)$ of nonnegative integers such that $f_i$ is the number of $i$-dimensional faces in the simplicial complex.

Since the $i$-dimensional faces of the simplex correspond to all subsets of vertices that have cardinality $i+1$, the entries in the f-vector of the simplex are all binomial coefficients.

 i1 : S = ZZ[x_0..x_6]; i2 : netList for n from -1 to 6 list fVector simplexComplex(n, S) +-+-+--+--+--+--+-+-+ o2 = |1| | | | | | | | +-+-+--+--+--+--+-+-+ |1|1| | | | | | | +-+-+--+--+--+--+-+-+ |1|2|1 | | | | | | +-+-+--+--+--+--+-+-+ |1|3|3 |1 | | | | | +-+-+--+--+--+--+-+-+ |1|4|6 |4 |1 | | | | +-+-+--+--+--+--+-+-+ |1|5|10|10|5 |1 | | | +-+-+--+--+--+--+-+-+ |1|6|15|20|15|6 |1| | +-+-+--+--+--+--+-+-+ |1|7|21|35|35|21|7|1| +-+-+--+--+--+--+-+-+ i3 : assert all(1..7, i -> (fVector simplexComplex(6,S))#i === binomial(7,i))

Our classic examples of abstract simplicial complexes illustrate more possibilities.

 i4 : S = ZZ[x_1..x_16]; i5 : fVector bartnetteSphereComplex S o5 = {1, 8, 27, 38, 19} o5 : List i6 : fVector bjornerComplex S o6 = {1, 6, 15, 11} o6 : List i7 : fVector dunceHatComplex S o7 = {1, 8, 24, 17} o7 : List i8 : fVector poincareSphereComplex S o8 = {1, 16, 106, 180, 90} o8 : List i9 : fVector rudinBallComplex S o9 = {1, 14, 66, 94, 41} o9 : List

There are two "trivial" simplicial complexes: the irrelevant complex has the empty set as a facet whereas the void complex has no faces. Every abstract simplicial complex other than the void complex has a unique face of dimension $-1$.

 i10 : irrelevant = simplicialComplex monomialIdeal gens S o10 = simplicialComplex | 1 | o10 : SimplicialComplex i11 : fVector irrelevant o11 = {1} o11 : List i12 : assert(fVector irrelevant === {1}) i13 : void = simplicialComplex monomialIdeal 1_S o13 = simplicialComplex 0 o13 : SimplicialComplex i14 : fVector void o14 = {0} o14 : List i15 : assert(fVector void === {0})

The f-vector is computed as the Hilbert function of the quotient of an exterior algebra by the corresponding Stanley–Reisner ideal.