A lattice path matroid is a transversal matroid corresponding to a pair of lattice paths on the plane. A matroid base polytope is the polytope whose vertices are the incidence vectors of the bases of the given matroid. In this talk, we study the facial structures of matroid base polytopes corresponding to lattice path matroids. In the case of a border strip, we show that all faces of a lattice path matroid polytope can be described by certain subsets of deletions, contractions, and direct sums. In particular, we express them in terms of the lattice path obtained from the border strip. Subsequently, the facial structures of a lattice path matroid polytope for a general case are explained in terms of certain tilings of skew shapes inside the given region.

Conway-Coxeter friezes have been around for almost 50 years, but the recent development of cluster algebras has brought them into new light. Recontextualizing and extending these friezes to SL_k-tilings gives rise to applications in physics and algebraic combinatorics. In this talk, we examine the correspondence between various types of SL₂-tilings and the geometry of the Farey graph.