On the Ehrhart Polynomial of Schubert Matroids

TL;DR

This paper derives formulas for lattice point counts in Schubert matroid polytopes, establishing Ehrhart positivity for various classes and confirming related conjectures, thus advancing understanding of their combinatorial and geometric properties.

Contribution

It provides a new formula for Ehrhart polynomials of Schubert matroids, proves Ehrhart positivity for several classes, and introduces notched rectangle matroids with conjectured positivity.

Findings

Ehrhart polynomials for Schubert, uniform, and minimal matroids are explicitly characterized.

All sparse paving Schubert matroids are Ehrhart positive, confirming Ferroni's conjecture.

Subfamilies of notched rectangle matroids are Ehrhart positive, with a conjecture for all such matroids.

Abstract

In this paper, we give a formula for the number of lattice points in the dilations of Schubert matroid polytopes. As applications, we obtain the Ehrhart polynomials of uniform and minimal matroids as special cases, and give a recursive formula for the Ehrhart polynomials of $(a,b)$-Catalan matroids. Ferroni showed that uniform and minimal matroids are Ehrhart positive. We show that all sparse paving Schubert matroids are Ehrhart positive and their Ehrhart polynomials are coefficient-wisely bounded by those of minimal and uniform matroids. This confirms a conjecture of Ferroni for the case of sparse paving Schubert matroids. Furthermore, we introduce notched rectangle matroids, which include minimal matroids, sparse paving Schubert matroids and panhandle matroids. We show that three subfamilies of notched rectangle matroids are Ehrhart positive, and conjecture that all notched rectangle matroids are Ehrhart positive.