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On Lattice Path Matroid Polytopes: Integer Points and Ehrhart Polynomial

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Abstract

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.

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Acknowledgements

The first author was partially supported by ANR Grants GATO ANR-16-CE40-0009-01 and CAPPS ANR-17-CE40-0018. The second author was supported by the Israel Science Foundation Grant No. 1452/15 and the European Research Council H2020 programme Grant No. 678765. The last two authors were partially supported by ECOS Nord Project M13M01.

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Authors and Affiliations

  1. Laboratoire d’Informatique Fondamentale, Faculté des Sciences de Luminy, Aix-Marseille Université and CNRS, 13288, Marseille Cedex 9, France

    Kolja Knauer

  2. Department of Computer Science, Faculty of Natural Sciences, Ben-Gurion University of the Negev, Beer Sheva, 84105, Israel

    Leonardo Martínez-Sandoval

  3. Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, Place Eugène Bataillon, 34095, Montpellier Cedex, France

    Jorge Luis Ramírez Alfonsín

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  1. Kolja Knauer
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  2. Leonardo Martínez-Sandoval
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  3. Jorge Luis Ramírez Alfonsín
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Correspondence to Kolja Knauer.

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Knauer, K., Martínez-Sandoval, L. & Ramírez Alfonsín, J.L. On Lattice Path Matroid Polytopes: Integer Points and Ehrhart Polynomial. Discrete Comput Geom 60, 698–719 (2018). https://doi.org/10.1007/s00454-018-9965-4

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  • DOI: https://doi.org/10.1007/s00454-018-9965-4

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