Integer point enumeration on independence polytopes and half-open hypersimplices

TL;DR

This paper studies the Ehrhart polynomials of independence matroid polytopes, proving positivity of coefficients by decomposing them into half-open hypersimplices, advancing understanding of their combinatorial and geometric properties.

Contribution

It introduces a novel decomposition of independence polytopes into half-open hypersimplices, establishing Ehrhart positivity for these structures.

Findings

Ehrhart polynomials of independence polytopes have positive coefficients.

Half-open hypersimplices are Ehrhart positive.

Polytope tiling with disjoint half-open hypersimplices.

Abstract

In this paper we investigate the Ehrhart Theory of the independence matroid polytope of uniform matroids. It is proved that these polytopes have an Ehrhart polynomial with positive coefficients. To do that, we prove that indeed all half-open-hypersimplices are Ehrhart positive, and tile disjointly our polytope using them.