# fVector -- the f-vector of a simplicial complex

## Synopsis

• Usage:
f = fVector D
• Inputs:
• Optional inputs:
• Flag => , default value false, the flag f-vector if the simplicial complex is properly defined over a multigraded ring.
• Outputs:
• f, such that f#i is the number of faces in D of dimension i for -1 <= i <= dim D or of squarefree degree i.

## Description

The pentagonal bipyramid has 7 vertices, 15 edges and 10 triangles.
 i1 : R = ZZ[a..g]; i2 : bipyramid = simplicialComplex monomialIdeal( a*g, b*d, b*e, c*e, c*f, d*f) o2 = | efg bfg deg cdg bcg aef abf ade acd abc | o2 : SimplicialComplex i3 : f = fVector bipyramid o3 = HashTable{-1 => 1} 0 => 7 1 => 15 2 => 10 o3 : HashTable i4 : f#0 o4 = 7 i5 : f#1 o5 = 15 i6 : f#2 o6 = 10
Every simplicial complex other than the void complex has a unique face of dimension -1.
 i7 : void = simplicialComplex monomialIdeal 1_R o7 = 0 o7 : SimplicialComplex i8 : fVector void o8 = HashTable{-1 => 0} o8 : HashTable
For a larger examp;le we consider the polarization of an artinian monomial ideal from section 3.2 in Miller-Sturmfels, Combinatorial Commutative Algebra.
 i9 : S = ZZ[x_1..x_4, y_1..y_4, z_1..z_4]; i10 : I = monomialIdeal(x_1*x_2*x_3*x_4, y_1*y_2*y_3*y_4, z_1*z_2*z_3*z_4, x_1*x_2*x_3*y_1*y_2*z_1, x_1*y_1*y_2*y_3*z_1*z_2, x_1*x_2*y_1*z_1*z_2*z_3); o10 : MonomialIdeal of S i11 : D = simplicialComplex I; i12 : fVector D o12 = HashTable{-1 => 1 } 0 => 12 1 => 66 2 => 220 3 => 492 4 => 768 5 => 837 6 => 609 7 => 264 8 => 51 o12 : HashTable
The boundary of the 3-dimensional cross-polytope is 3-colorable. If we define this simplicial complex over a Z^3-graded ring we can ask for its flag f-vector.
 i13 : grading = {{1,0,0},{1,0,0},{0,1,0},{0,1,0},{0,0,1},{0,0,1}}; i14 : S = ZZ[x_1..x_6, Degrees => grading]; i15 : I = monomialIdeal(x_1*x_2,x_3*x_4,x_5*x_6); o15 : MonomialIdeal of S i16 : fVector simplicialComplex I o16 = HashTable{-1 => 1} 0 => 6 1 => 12 2 => 8 o16 : HashTable i17 : fVector(simplicialComplex I, Flag => true) o17 = HashTable{{0, 0, 0} => 1} {0, 0, 1} => 2 {0, 1, 0} => 2 {0, 1, 1} => 4 {1, 0, 0} => 2 {1, 0, 1} => 4 {1, 1, 0} => 4 {1, 1, 1} => 8 o17 : HashTable

The f-vector is computed using the Hilbert series of the Stanley-Reisner ideal. For example, see Hosten and Smith's chapter Monomial Ideals, in Computations in Algebraic Geometry with Macaulay2, Springer 2001.

## Caveat

The option Flag checks if the multigrading corresponds to a properly d-coloring of D, where d is the dimension of D plus one. If it is not the case the output is an empty HashTable.