Lattice Path Matroid Polytopes

TL;DR

This paper studies the geometric and combinatorial properties of lattice path matroid polytopes, including their face structure, decompositions, triangulations, Ehrhart polynomials, and volume, linking lattice paths with matroid theory.

Contribution

It introduces and analyzes lattice path matroid polytopes, exploring their structural properties and providing new insights into their geometric and combinatorial characteristics.

Findings

Characterization of face structures of lattice path matroid polytopes

Results on decompositions and triangulations of these polytopes

Formulas and properties of Ehrhart polynomials and volume

Abstract

Fix two lattice paths $P$ and $Q$ from $(0,0)$ to $(m,r)$ that use East and North steps with $P $ never going above $Q$. Bonin et al. show that the lattice paths that go from $(0,0)$ to $(m,r)$ and remain bounded by $P$ and $Q$ can be identified with the bases of a particular type of transversal matroid, which we call it a lattice path matroid. In this paper, we consider properties of lattice path matroid polytopes. These are the polytopes associated to the lattice path matroids. We investigate their face structure, decomposition, triangulation, Ehrhart polynomial and volume.