  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.
