# On lattice path matroid polytopes: integer points and Ehrhart polynomial

## TL;DR

This paper studies lattice path matroid polytopes, characterizing their integer points, establishing affine equivalences, and deriving Ehrhart polynomial formulas, advancing understanding of their combinatorial and geometric properties.

## Contribution

It provides a characterization of integer points in lattice path matroid polytopes and derives explicit Ehrhart polynomial formulas, linking these polytopes to distributive polytopes.

## Key findings

- Integer points characterized as polygonal paths
- Lattice path matroid polytopes are affinely equivalent to distributive polytopes
- Explicit Ehrhart polynomial formula for a specific family

## Abstract

In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove that lattice path matroid polytopes are affinely equivalent to a family of distributive polytopes. As applications we obtain two new infinite families of matroids verifying a conjecture of De Loera et.~al. and present an explicit formula of the Ehrhart polynomial for one of them.