Lattice points in slices of prisms

TL;DR

This paper provides a combinatorial framework for understanding the Ehrhart theory of slices of prisms, generalizing hypersimplices, and establishes Ehrhart positivity along with interpretations of related polynomials.

Contribution

It introduces a combinatorial formula for Ehrhart coefficients of these polytopes, proving their Ehrhart positivity and extending interpretations of the $h^*$-polynomial coefficients.

Findings

Ehrhart coefficients expressed via weighted permutations.

Proof of Ehrhart positivity for these polytopes.

Combinatorial interpretation of the $h^*$-polynomial coefficients.

Abstract

We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov; moreover, they coincide with polymatroids satisfying the strong exchange property up to an affinity. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive. Additionally, via an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the $h^*$-polynomial. All of our results provide a combinatorial understanding of the Hilbert functions and the $h$-vectors of all algebras of Veronese type, a problem that had remained elusive up to this point. A variety of applications are discussed, including expressions for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; some extensions of Laplace's result on the combinatorial interpretation of the volume of the hypersimplex; a multivariate generalization of the flag Eulerian numbers and refinements; and a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.