H-vector

Recurrence relation

The ${\displaystyle h}$-vector ${\displaystyle (h_0,h_1,\dots ,h_d)}$ can be computed from the ${\displaystyle f}$-vector ${\displaystyle (f_{-1},f_0,\dots ,f_{d-1})}$ by using the recurrence relation

${\displaystyle h_0^i=1,-1\le i\le d}$

${\displaystyle h_{i+1}^i=f_i,-1\le i\le d-1}$

${\displaystyle h_k^i=h_k^{i-1}-h_{k-1}^{i-1},1\le k\le i\le d}$.

and finally setting ${\displaystyle h_k=h_k^d}$ for ${\displaystyle 0\le k\le d}$. For small examples, one can use this method to compute ${\displaystyle h}$-vectors quickly by hand by recursively filling the entries of an array similar to Pascal's triangle. For example, consider the boundary complex ${\displaystyle \mathrm{\Delta }}$ of an octahedron. The ${\displaystyle f}$-vector of ${\displaystyle \mathrm{\Delta }}$ is ${\displaystyle (1,6,12,8)}$. To compute the ${\displaystyle h}$-vector of ${\displaystyle \mathrm{\Delta }}$, construct a triangular array by first writing ${\displaystyle d+2}$ ${\displaystyle 1}$s down the left edge and the ${\displaystyle f}$-vector down the right edge.

${\displaystyle \begin{array}{cccccccc}& & & & 1& & & \\ & & & 1& & 6& & \\ & & 1& & & & 12& \\ & 1& & & & & & 8\\ 1& & & & & & & & 0\end{array}}$

(We set ${\displaystyle f_d=0}$ just to make the array triangular.) Then, starting from the top, fill each remaining entry by subtracting its upper-left neighbor from its upper-right neighbor. In this way, we generate the following array:

${\displaystyle \begin{array}{cccccccc}& & & & 1& & & \\ & & & 1& & 6& & \\ & & 1& & 5& & 12& \\ & 1& & 4& & 7& & 8\\ 1& & 3& & 3& & 1& & 0\end{array}}$

The entries of the bottom row (apart from the final ${\displaystyle 0}$) are the entries of the ${\displaystyle h}$-vector. Hence, the ${\displaystyle h}$-vector of ${\displaystyle \mathrm{\Delta }}$ is ${\displaystyle (1,3,3,1)}$.

Toric h-vector

To an arbitrary graded poset P, Stanley associated a pair of polynomials f(P,x) and g(P,x). Their definition is recursive in terms of the polynomials associated to intervals [0,y] for all y ∈ P, y ≠ 1, viewed as ranked posets of lower rank (0 and 1 denote the minimal and the maximal elements of P). The coefficients of f(P,x) form the toric h-vector of P. When P is an Eulerian poset of rank d + 1 such that P − 1 is simplicial, the toric h-vector coincides with the ordinary h-vector constructed using the numbers f_i of elements of P − 1 of given rank i + 1. In this case the toric h-vector of P satisfies the Dehn–Sommerville equations

${\displaystyle h_k=h_{d-k}.}$

The reason for the adjective "toric" is a connection of the toric h-vector with the intersection cohomology of a certain projective toric variety X whenever P is the boundary complex of rational convex polytope. Namely, the components are the dimensions of the even intersection cohomology groups of X:

${\displaystyle h_k={\dim }_{\mathbb{Q}}{\mathrm{IH}}^{2k}(X,\mathbb{Q})}$

(the odd intersection cohomology groups of X are all zero). The Dehn–Sommerville equations are a manifestation of the Poincaré duality in the intersection cohomology of X. Kalle Karu proved that the toric h-vector of a polytope is unimodal, regardless of whether the polytope is rational or not.

Flag h-vector and cd-index

A different generalization of the notions of f-vector and h-vector of a convex polytope has been extensively studied. Let ${\displaystyle P}$ be a finite graded poset of rank n, so that each maximal chain in ${\displaystyle P}$ has length n. For any ${\displaystyle S}$, a subset of ${\displaystyle \left\{0,\dots ,n\right\}}$, let ${\displaystyle {\alpha }_P(S)}$ denote the number of chains in ${\displaystyle P}$ whose ranks constitute the set ${\displaystyle S}$. More formally, let

${\displaystyle rk:P\to \{0,1,\dots ,n\}}$

be the rank function of ${\displaystyle P}$ and let ${\displaystyle P_S}$ be the ${\displaystyle S}$-rank selected subposet, which consists of the elements from ${\displaystyle P}$ whose rank is in ${\displaystyle S}$:

${\displaystyle P_S=\{x\in P:rk(x)\in S\}.}$

Then ${\displaystyle {\alpha }_P(S)}$ is the number of the maximal chains in ${\displaystyle P_S}$ and the function

${\displaystyle S\mapsto {\alpha }_P(S)}$

is called the flag f-vector of P. The function

${\displaystyle S\mapsto {\beta }_P(S),{\beta }_P(S)=\sum _{T\subseteq S}(-1{)}^{|S|-|T|}{\alpha }_P(S)}$

is called the flag h-vector of ${\displaystyle P}$. By the inclusion–exclusion principle,

${\displaystyle {\alpha }_P(S)=\sum _{T\subseteq S}{\beta }_P(T).}$

The flag f- and h-vectors of ${\displaystyle P}$ refine the ordinary f- and h-vectors of its order complex ${\displaystyle \mathrm{\Delta }(P)}$:

${\displaystyle f_{i-1}(\mathrm{\Delta }(P))=\sum _{|S|=i}{\alpha }_P(S),h_i(\mathrm{\Delta }(P))=\sum _{|S|=i}{\beta }_P(S).}$

The flag h-vector of ${\displaystyle P}$ can be displayed via a polynomial in noncommutative variables a and b. For any subset ${\displaystyle S}$ of {1,…,n}, define the corresponding monomial in a and b,

${\displaystyle u_S=u_1\cdots u_n,u_i=a\text{ for }i\notin S,u_i=b\text{ for }i\in S.}$

Then the noncommutative generating function for the flag h-vector of P is defined by

${\displaystyle {\mathrm{\Psi }}_P(a,b)=\sum _S{\beta }_P(S)u_S.}$

From the relation between α_P(S) and β_P(S), the noncommutative generating function for the flag f-vector of P is

${\displaystyle {\mathrm{\Psi }}_P(a,a+b)=\sum _S{\alpha }_P(S)u_S.}$

Margaret Bayer and Louis Billera determined the most general linear relations that hold between the components of the flag h-vector of an Eulerian poset P.

Fine noted an elegant way to state these relations: there exists a noncommutative polynomial Φ_P(c,d), called the cd-index of P, such that

${\displaystyle {\mathrm{\Psi }}_P(a,b)={\mathrm{\Phi }}_P(a+b,ab+ba).}$

Stanley proved that all coefficients of the cd-index of the boundary complex of a convex polytope are non-negative. He conjectured that this positivity phenomenon persists for a more general class of Eulerian posets that Stanley calls Gorenstein* complexes and which includes simplicial spheres and complete fans. This conjecture was proved by Kalle Karu. The combinatorial meaning of these non-negative coefficients (an answer to the question "what do they count?") remains unclear.