The Ehrhart $h^*$-polynomials of positroid polytopes

TL;DR

This paper provides explicit formulas for the Ehrhart $h^*$-polynomials of positroid polytopes, connecting combinatorial properties of permutations with geometric lattice point enumeration.

Contribution

It generalizes previous results on hypersimplices by deriving formulas for $h^*$-polynomials of all positroid polytopes based on permutation descents.

Findings

Explicit formulas for $h^*$-polynomials of positroid polytopes

Connection between permutation descents and Ehrhart series

Generalization of hypersimplex results

Abstract

A positroid is a matroid realized by a matrix such that all maximal minors are non-negative. Positroid polytopes are matroid polytopes of positroids. In particular, they are lattice polytopes. The Ehrhart polynomial of a lattice polytope counts the number of integer points in the dilation of that polytope. The Ehrhart series is the generating function of the Ehrhart polynomial, a rational function with a numerator called the $h^*$-polynomial. We give explicit formulas for the $h^*$-polynomials of an arbitrary positroid polytope regarding permutation descents. Our result generalizes that of Early, Kim, and Li for hypersimplices.