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 stated that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes. It was proven in December 2018 by Karim Adiprasito. 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.
Definition Let Δ be an abstract simplicial complex of dimension d − 1 with fi i-dimensional faces and f−1 = 1. These numbers are arranged into the f-vector of Δ,
f ( Δ ) = ( f − 1 , f 0 , … , f d − 1 ) . {\displaystyle f(\Delta )=(f_{-1},f_{0},\ldots ,f_{d-1}).}
An important special case occurs when Δ is the boundary of a d-dimensional convex polytope. For k = 0, 1, …, d, let
h k = ∑ i = 0 k ( − 1 ) k − i ( d − i k − i ) f i − 1 . {\displaystyle h_{k}=\sum _{i=0}^{k}(-1)^{k-i}{\binom {d-i}{k-i}}f_{i-1}.}
The tuple
h ( Δ ) = ( h 0 , h 1 , … , h d ) {\displaystyle h(\Delta )=(h_{0},h_{1},\ldots ,h_{d})}
is called the h-vector of Δ. In particular, h 0 = 1 {\displaystyle h_{0}=1} , h 1 = f 0 − d {\displaystyle h_{1}=f_{0}-d} , and h d = ( − 1 ) d ( 1 − χ ( Δ ) ) {\displaystyle h_{d}=(-1)^{d}(1-\chi (\Delta ))} , where χ ( Δ ) {\displaystyle \chi (\Delta )} is the Euler characteristic of Δ {\displaystyle \Delta } . The f-vector and the h-vector uniquely determine each other through the linear relation
∑ i = 0 d f i − 1 ( t − 1 ) d − i = ∑ k = 0 d h k t d − k , {\displaystyle \sum _{i=0}^{d}f_{i-1}(t-1)^{d-i}=\sum _{k=0}^{d}h_{k}t^{d-k},}
from which it follows that, for i = 0 , … , d {\displaystyle i=0,\dotsc ,d} ,
f i − 1 = ∑ k = 0 i ( d − k i − k ) h k . {\displaystyle f_{i-1}=\sum _{k=0}^{i}{\binom {d-k}{i-k}}h_{k}.}
In particular, f d − 1 = h 0 + h 1 + ⋯ + h d {\displaystyle f_{d-1}=h_{0}+h_{1}+\dotsb +h_{d}} . Let R = k[Δ] be the Stanley–Reisner ring of Δ. Then its Hilbert–Poincaré series can be expressed as
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