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.[citation needed] 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
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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. In particular, , , and , where is the Euler characteristic of . The f-vector and the h-vector uniquely determine each other through the linear relation
The h-vector is closely related to the h*-vector for a convex lattice polytope, see Ehrhart polynomial.
Recurrence relation
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The -vector can be computed from the -vector by using the recurrence relation
.
and finally setting for . For small examples, one can use this method to compute -vectors quickly by hand by recursively filling the entries of an array similar to Pascal's triangle. For example, consider the boundary complex of an octahedron. The -vector of is . To compute the -vector of , construct a triangular array by first writing s down the left edge and the -vector down the right edge.
(We set 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:
The entries of the bottom row (apart from the final ) are the entries of the -vector. Hence, the -vector of is .
Toric h-vector
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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 fi of elements of P − 1 of given rank i + 1. In this case the toric h-vector of P satisfies the Dehn–Sommerville equations
The reason for the adjective "toric" is a connection of the toric h-vector with the intersection cohomology of a certain projectivetoric varietyX whenever P is the boundary complex of rational convex polytope. Namely, the components are the dimensions of the even intersection cohomology groups of X:
(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.[7]
Flag h-vector and cd-index
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A different generalization of the notions of f-vector and h-vector of a convex polytope has been extensively studied. Let be a finite graded poset of rank n, so that each maximal chain in has length n. For any , a subset of , let denote the number of chains in whose ranks constitute the set . More formally, let
be the rank function of and let be the -rank selected subposet, which consists of the elements from whose rank is in :
Then is the number of the maximal chains in and the function
The flag f- and h-vectors of refine the ordinary f- and h-vectors of its order complex:[8]
The flag h-vector of can be displayed via a polynomial in noncommutative variables a and b. For any subset of {1,…,n}, define the corresponding monomial in a and b,
Then the noncommutative generating function for the flag h-vector of P is defined by
From the relation between αP(S) and βP(S), the noncommutative generating function for the flag f-vector of P is
Fine noted an elegant way to state these relations: there exists a noncommutative polynomial ΦP(c,d), called the cd-index of P, such that
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.[10] The combinatorial meaning of these non-negative coefficients (an answer to the question "what do they count?") remains unclear.
^Billera, Louis; Lee, Carl (1981), "A proof of the sufficiency of McMullen's conditions for f-vectors of simplicial convex polytopes", Journal of Combinatorial Theory, Series A, 31 (3): 237–255, doi:10.1016/0097-3165(81)90058-3.
^Bayer, Margaret M. and Billera, Louis J (1985), "Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets", Inventiones Mathematicae 79: 143-158. doi:10.1007/BF01388660.
^Karu, Kalle (2006), "The cd-index of fans and posets", Compositio Mathematica, 142 (3): 701–718, doi:10.1112/S0010437X06001928, MR 2231198.
Further reading
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Stanley, Richard (1996), Combinatorics and commutative algebra, Progress in Mathematics, vol. 41 (2nd ed.), Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3836-9.
Stanley, Richard (1997), Enumerative Combinatorics, vol. 1, Cambridge University Press, ISBN 0-521-55309-1.