H-vector
In algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Dehn–Sommerville equations in a particularly simple form. A characterization of the set of h-vectors of simplicial polytopes was conjectured by Peter McMullen and proved by Lou Billera and Carl W. Lee and Richard Stanley (g-theorem). The definition of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture states that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes.
Stanley introduced a generalization of the h-vector, the toric h-vector, which is defined for an arbitrary ranked poset, and proved that for the class of Eulerian posets, the Dehn–Sommerville equations continue to hold. A different, more combinatorial, generalization of the h-vector that has been extensively studied is the flag h-vector of a ranked poset. For Eulerian posets, it can be more concisely expressed by means of a noncommutative polynomial in two variables called the cd-index.
Let Δ be an abstract simplicial complex of dimension d − 1 with fii-dimensional faces and f−1 = 1. These numbers are arranged into the f-vector of Δ,
An important special case occurs when Δ is the boundary of a d-dimensional convex polytope.
For k = 0, 1, …, d, let
The tuple
is called the h-vector of Δ. The f-vector and the h-vector uniquely determine each other through the linear relation
Let R = k[Δ] be the Stanley–Reisner ring of Δ. Then its Hilbert–Poincaré series can be expressed as
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