Shelling and the h-Vector of the (Extra)ordinary Polytope

Summary

Ordinary polytopes were introduced by Bisztriczky as a (nonsimplicial) generalization of cyclic polytopes. We show that the colex order of facets of the ordinary polytope is a shelling order. This shelling shares many nice properties with the shellings of simplicial polytopes. We also give a shallow triangulation of the ordinary polytope, and show how the shelling and the triangulation are used to compute the toric h-vector of the ordinary polytope. As one consequence, we get that the contribution from each shelling component to the h-vector is nonnegative. Another consequence is a combinatorial proof that the entries of the h-vector of any ordinary polytope are simple sums of binomial coefficients.

1. Introduction

This paper has a couple of main motivations. The first comes from the study of toric h-vectors of convex polytopes. The h-vector played a crucial role in the characterization of face vectors of simplicial polytopes [Billera and Lee 1981; McMullen and Shephard 1971; Stanley 1980]. In the simplicial case, the h-vector is linearly equivalent to the face vector, and has a combinatorial interpretation in a shelling of the polytope. The h -vector of a simplicial polytope is also the sequence of Betti numbers of an associated toric variety. In this context it generalizes to nonsimplicial polytopes. However, for nonsimplicial polytopes, we do not have a good combinatorial understanding of the entries of the h -vector. (Chan [1991] gives a combinatorial interpretation for the h-vector of cubical polytopes.)

The definition of the (toric) h-vector for general polytopes (and even more generally, for Eulerian posets) first appeared in [Stanley 1987]. Already there Stanley raised the issue of computing the h-vector from a shelling of the polytope.