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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:

Zbl 1207.52015
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References:

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