Editing Simplicial complex (section)

== Combinatorics ==
[[Combinatorics|Combinatorialists]] often study the '''''f''-vector''' of a simplicial d-complex Δ, which is the [[integer]] sequence <math>(f_0, f_1, f_2, \ldots, f_{d+1})</math>, where ''f''<sub>''i''</sub> is the number of (''i''−1)-dimensional faces of Δ (by convention, ''f''<sub>0</sub>&nbsp;=&nbsp;1 unless Δ is the empty complex). For instance, if Δ is the boundary of the [[octahedron]], then its ''f''-vector is (1, 6, 12, 8), and if Δ is the first simplicial complex pictured above, its ''f''-vector is (1, 18, 23, 8, 1). A complete characterization of the possible ''f''-vectors of simplicial complexes is given by the [[Kruskal–Katona theorem]].

By using the ''f''-vector of a simplicial ''d''-complex Δ as coefficients of a [[polynomial]] (written in decreasing order of exponents), we obtain the '''f-polynomial''' of Δ. In our two examples above, the ''f''-polynomials would be <math>x^3+6x^2+12x+8</math> and <math>x^4+18x^3+23x^2+8x+1</math>, respectively.

Combinatorists are often quite interested in the '''h-vector''' of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging ''x''&nbsp;−&nbsp;1 into the ''f''-polynomial of Δ. Formally, if we write ''F''<sub>Δ</sub>(''x'') to mean the ''f''-polynomial of Δ, then the '''h-polynomial''' of Δ is
: <math>F_\Delta(x-1)=h_0x^{d+1}+h_1x^d+h_2x^{d-1}+\cdots+h_dx+h_{d+1}</math>
and the ''h''-vector of Δ is
: <math>(h_0,  h_1,  h_2, \cdots,  h_{d+1}).</math>

We calculate the h-vector of the octahedron boundary (our first example) as follows:
: <math>F(x-1)=(x-1)^3+6(x-1)^2+12(x-1)+8=x^3+3x^2+3x+1.</math>

So the ''h''-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this ''h''-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicial [[polytope]] (these are the [[Dehn–Sommerville equations]]). In general, however, the ''h''-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting ''h''-vector is (1, 3, −2).

A complete characterization of all simplicial polytope ''h''-vectors is given by the celebrated [[g-theorem]] of [[Richard P. Stanley|Stanley]], Billera, and Lee.

Simplicial complexes can be seen to have the same geometric structure as the [[contact graph]] of a [[sphere packing]] (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of sphere packings, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.