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# flagfVector(List,SimplicialComplex) -- compute a flag $f$-number of a colored simplicial complex

## Synopsis

• Function: flagfVector
• Usage:
f = flagfVector(L, Delta)
• Inputs:
• L, a list,
• Delta, ,
• Outputs:
• an integer, the flag f-number corresponding to the color set determined by L

## Description

A coloring of an abstract simplicial complex $\Delta$ is a labelling of its vertices with colors. The color set of a face is the set of colors of its vertices. Given a set of colors $L$, the flag $f$-number $f_L(\Delta)$ is the number of faces with color set $L$. In this package, a coloring of an abstract simplicial complex is determined by a multigrading of its ambient ring.

If we color the $2$-simplex with distinct colors, each color set corresponds to a unique face.

 i1 : S = QQ[a,b,c, DegreeRank => 3]; i2 : degrees S o2 = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}} o2 : List i3 : Δ = simplexComplex(2, S) o3 = simplicialComplex | abc | o3 : SimplicialComplex i4 : flagfVector({0,0,0}, Δ) o4 = 1 i5 : flagfVector({1,0,0}, Δ) o5 = 1 i6 : flagfVector({0,1,0}, Δ) o6 = 1 i7 : flagfVector({0,0,1}, Δ) o7 = 1 i8 : flagfVector({0,1,1}, Δ) o8 = 1 i9 : flagfVector({1,0,1}, Δ) o9 = 1 i10 : flagfVector({1,1,0}, Δ) o10 = 1 i11 : flagfVector({1,1,1}, Δ) o11 = 1

A coloring is proper if no two vertices in the same face have the same color. The bowtie complex has a proper $3$-coloring.

 i12 : R = ZZ[a..e, Degrees => {{1,0,0},{0,1,0},{0,0,1},{1,0,0},{0,1,0}}]; i13 : Γ = simplicialComplex {a*b*c, c*d*e} o13 = simplicialComplex | cde abc | o13 : SimplicialComplex i14 : assert isProper Γ i15 : flagfVector({0,0,0}, Γ) o15 = 1 i16 : flagfVector({1,0,0}, Γ) o16 = 2 i17 : flagfVector({0,1,0}, Γ) o17 = 2 i18 : flagfVector({0,0,1}, Γ) o18 = 1 i19 : flagfVector({0,1,1}, Γ) o19 = 2 i20 : flagfVector({1,0,1}, Γ) o20 = 2 i21 : flagfVector({1,1,0}, Γ) o21 = 2 i22 : flagfVector({1,1,1}, Γ) o22 = 2

The method function $\operatorname{flagfVector}$ does not check whether the multigrading on ambient ring determines a proper coloring.

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

## Caveat

Not every grading of the ambient polynomial ring corresponds to a coloring.