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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. arXiv:1701.05529v2 [math.CO]

Abstract. Let M be a matroid without loops or coloops and let T (M ; x, y) be its Tutte polynomial. In 1999 Merino and Welsh conjectured that max(T (M ; 2, 0), T (M ; 0, 2)) ≥ T (M ; 1, 1) holds for graphic matroids. Ten years later, Conde and Merino proposed a multiplicative version of the conjecture which implies the original one. In this paper we prove the multiplicative conjecture for the family of lattice path matroids (generalizing earlier results on uniform and Catalan matroids). In order to do this, we introduce and study particular lattice path matroids, called snakes, used as building bricks to indeed establish a strengthening of the multiplicative conjecture as well as a complete characterization of the cases in which equality holds.

Let P (M) be the matroid base polytope of a matroid M. A matroid base polytope decomposition of P (M) is a decomposition of the formis also a matroid base polytope for some matroid M i , and for each 1In this paper, we investigate hyperplane splits, that is, polytope decompositions when t = 2. We give sufficient conditions for M so P (M) has a hyperplane split and characterize when P (M 1 ⊕ M 2 ) has a hyperplane split where M 1 ⊕ M 2 denote the direct sum of matroids M 1 and M 2 . We also prove that P (M) has not a hyperplane split if M is binary. Finally, we show that P (M)has not a decomposition if its 1-skeleton is the hypercube.