On lattice path matroid polytopes: alcoved triangulations and snake decompositions

Abstract

We study lattice path matroid polytopes using their alcoved triangulation. We characterize Gorenstein lattice path matroid polytopes, yielding a new class of matroids satisfying the unimodality conjecture of de Loera, Haws, and K{\"o}ppe. Further, we characterize matroids whose polytopes are order polytopes as a special class of lattice path matroids, called snakes. Finally, we give combinatorial interpretations of the volumes and $h^*$-vectors of lattice path matroids of rank $2$ based on their snake decomposition.