Lattice Path Delta Matroids

Abstract

We initiate the study of a type $C_n$ generalization of the lattice path matroids defined by Bonin, de Mier, and Noy. These are delta matroids whose feasible sets are in bijection with lattice paths which are symmetric along the main diagonal. We describe deletion, contraction, enveloping matroids, and the homogeneous components of these delta matroids in terms of the combinatorics of symmetric lattice paths. In the second half, we study the convex geometry of the feasible polytopes of lattice path delta matroids. We show that these polytopes decompose into the feasible polytope of special lattice paths corresponding to intervals in the type $C_n$ Gale order whose related Richardson variety is a toric variety. Further, these special polytopes inherit a unimodular triangulation from Stanley's Eulerian triangulation of the hypercube. As a consequence, the volume of these polytopes is the proportion of permutations with specific ascent sets.