Lattice points in slices of prisms

Luis Ferroni, Daniel McGinnis

Published 2022-02-23Version 1

We study the Ehrhart polynomials of certain slices of rectangular prisms. These polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov. 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, inspired by a result due to Early and Kim, we give a combinatorial interpretation for all the coefficients of the $h^*$-polynomial; this solves the problem of providing an interpretation for the numerator of the Hilbert series, also known as the $h$-vector of all algebras of Veronese type. As corollaries of our results, we obtain an expression for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; we use this to provide some generalizations of Laplace's result on the combinatorial interpretation of the volume of the hypersimplex. We discuss an application regarding a generalization of the flag Eulerian numbers and certain refinements, and give a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.