On Lattice Path Matroid Polytopes: Integer Points and Ehrhart Polynomial
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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.
KeywordsEhrhart polynomial Distributive polytope Matroid base polytope Lattice path matroid
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.
- 1.An, S., Jung, J., Kim, S.: Facial structures of lattice path matroid polytopes. arXiv:1701.00362 (2017)
- 2.Bidkhori, H.: Lattice path matroid polytopes. arXiv:1212.5705 (2012)
- 9.Cohen, E., Tetali, P., Yeliussizov, D.: Lattice path matroids: negative correlation and fast mixing. arXiv:1505.06710 (2015)
- 21.Oxley, J.: Matroid Theory. Oxford Graduate Texts in Mathematics, vol. 21, 2nd edn. Oxford University Press, Oxford (2011)Google Scholar
- 26.Stanley, R.P.: A chromatic-like polynomial for ordered sets. In: Proceedings of the 2nd Chapel Hill Conference on Combinatorial Mathematics and its Applications, pp. 421–427. University of North Carolina, Chapel Hill (1970)Google Scholar
- 31.Welsh, D.J.A.: Matroid Theory. L. M. S. Monographs, vol. 8. Academic Press, London (1976)Google Scholar