Ehrhart positivity for lattice path matroids

TL;DR

This paper proves that all lattice path matroids are Ehrhart positive, unifying previous results and supporting several conjectures in the field of matroid theory and Ehrhart polynomials.

Contribution

It establishes Ehrhart positivity for all lattice path matroids, confirming multiple longstanding conjectures and extending prior work on matroid Ehrhart properties.

Findings

All lattice path matroids are Ehrhart positive.

Supports conjecture on Ehrhart positivity of positroids.

Implicates Ehrhart positivity of Schubert matroids.

Abstract

We prove that all lattice path matroids are Ehrhart positive. This unifies and generalizes numerous results on the Ehrhart positivity of matroids developed over the last two decades. We rely on our previous work on the positivity of order polynomials of fences. Our main result supports the conjecture by Ferroni, Jochemko, and Schr\"oter (2022) on the Ehrhart positivity of positroids. Furthermore, our main result implies that all Schubert matroids are Ehrhart positive, which thus settles a conjecture by Fan and Li (2024), and supports a conjecture by Monical, Tokcan, and Yong (2019) on the Ehrhart positivity of Schubitopes.