$h^*$-Vectors, Eulerian Polynomials and Stable Polytopes of Graphs

h*-Vectors, Eulerian Polynomials and Stable Polytopes of Graphs

Conditions are given on a lattice polytope P of dimension m or its associated affine semigroup ring which imply inequalities for the h∗-vector (h∗0, h∗1, . . . , h∗m) of P of the form hi ≥ hd−i for 1 ≤ i ≤ bd/2c and hbd/2c ≥ hbd/2c+1 ≥ · · · ≥ hd, where hi = 0 for d < i ≤ m. Two applications to order polytopes of posets and stable polytopes of perfect graphs are included.

h-Vectors of Gorenstein polytopes

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a unimodular triangulation satisfies McMullen’s g-theorem; in particular it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a “long” re...

Rational lecture hall polytopes and inflated Eulerian polynomials

For a sequence of positive integers s = (s1, . . . , sn), we define the rational lecture hall polytope R n . We prove that its h∗-polynomial, Q (s) n (x), has nonnegative integer coefficients that count certain statistics on s-inversion sequences. The polynomial Q n (x) can be viewed as an inflated version of the s-Eulerian polynomial, A n (x), associated with the integral lecture hall polytope...

Matroids: h-vectors, Zonotopes, and Lawrence Polytopes

Stable multivariate Eulerian polynomials and generalized Stirling permutations

We study Eulerian polynomials as the generating polynomials of the descent statistic over Stirling permutations – a class of restricted multiset permutations. We develop their multivariate refinements by indexing variables by the values at the descent tops, rather than the position where they appear. We prove that the obtained multivariate polynomials are stable, in the sense that they do not v...