On lattice path matroid polytopes: integer points and Ehrhart polynomial. (English) Zbl 1494.52014
Summary: 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 J. A. De Loera et. al. [Discrete Comput. Geom. 42, No. 4, 670–704 (2009; Zbl 1207.52015)] and present an explicit formula of the Ehrhart polynomial for one of them.
MSC:
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
Citations:
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