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fVector -- the f-vector of a simplicial complex

Synopsis

Description

loadPackage "SimplicialComplexes";
The pentagonal bipyramid has 7 vertices, 15 edges and 10 triangles.
R = ZZ[a..g];
bipyramid = simplicialComplex monomialIdeal( a*g, b*d, b*e, c*e, c*f, d*f)
f = fVector bipyramid
f#0
f#1
f#2
Every simplicial complex other than the void complex has a unique face of dimension -1.
void = simplicialComplex monomialIdeal 1_R
fVector void
For a larger examp;le we consider the polarization of an artinian monomial ideal from section 3.2 in Miller-Sturmfels, Combinatorial Commutative Algebra.
S = ZZ[x_1..x_4, y_1..y_4, z_1..z_4];
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);
D = simplicialComplex I;
fVector D

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.

See also

Ways to use fVector :