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Discrete & Computational Geometry

, Volume 60, Issue 3, pp 698–719 | Cite as

On Lattice Path Matroid Polytopes: Integer Points and Ehrhart Polynomial

  • Kolja KnauerEmail author
  • Leonardo Martínez-Sandoval
  • Jorge Luis Ramírez Alfonsín
Article

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.

Keywords

Ehrhart polynomial Distributive polytope Matroid base polytope Lattice path matroid 

Notes

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.

References

  1. 1.
    An, S., Jung, J., Kim, S.: Facial structures of lattice path matroid polytopes. arXiv:1701.00362 (2017)
  2. 2.
    Bidkhori, H.: Lattice path matroid polytopes. arXiv:1212.5705 (2012)
  3. 3.
    Birkhoff, G.: Rings of sets. Duke Math. J. 3(3), 443–454 (1937)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonin, J.E.: Lattice path matroids: the excluded minors. J. Combin. Theory Ser. B 100(6), 585–599 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonin, J., de Mier, A., Noy, M.: Lattice path matroids: enumerative aspects and Tutte polynomials. J. Combin. Theory Ser. A 104(1), 63–94 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonin, J.E., Giménez, O.: Multi-path matroids. Combin. Probab. Comput. 16(2), 193–217 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brändén, P.: Unimodality, log-concavity, real-rootedness and beyond. In: Bona, M. (ed.) Handbook of Enumerative Combinatorics. Discrete Mathematics and Its Applications, pp. 437–483. CRC Press, Boca Raton (2015)CrossRefGoogle Scholar
  8. 8.
    Chatelain, V., Ramírez Alfonsín, J.L.: Matroid base polytope decomposition. Adv. Appl. Math. 47(1), 158–172 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cohen, E., Tetali, P., Yeliussizov, D.: Lattice path matroids: negative correlation and fast mixing. arXiv:1505.06710 (2015)
  10. 10.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, New York (2002)CrossRefzbMATHGoogle Scholar
  11. 11.
    De Loera, J.A., Haws, D.C., Köppe, M.: Ehrhart polynomials of matroid polytopes and polymatroids. Discrete Comput. Geom. 42(4), 670–702 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Delucchi, E., Dlugosch, M.: Bergman complexes of lattice path matroids. SIAM J. Discrete Math. 29(4), 1916–1930 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. (2) 51(1), 161–166 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ehrhart, E.: Sur les polyèdres rationnels homothétiques à \(n\) dimensions. C. R. Acad. Sci. 254, 616–618 (1962)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Feichtner, E.M., Sturmfels, B.: Matroid polytopes, nested sets and Bergman fans. Port. Math. (N.S.) 62(4), 437–468 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Felsner, S., Knauer, K.: Distributive lattices, polyhedra, and generalized flows. Eur. J. Combin. 32(1), 45–59 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Katzman, M.: The Hilbert series of Veronese type. Commun. Algebra 33(4), 1141–1146 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Knauer, K., Martínez-Sandoval, L., Alfonsín, J.L.: A Tutte polynomial inequality for lattice path matroids. Adv. Appl. Math. 94, 23–38 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Morton, J., Turner, J.: Computing the Tutte polynomial of lattice path matroids using determinantal circuits. Theor. Comput. Sci. 598, 150–156 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Neggers, J.: Representations of finite partially ordered sets. J. Combin. Inf. Syst. Sci. 3(3), 113–133 (1978)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Oxley, J.: Matroid Theory. Oxford Graduate Texts in Mathematics, vol. 21, 2nd edn. Oxford University Press, Oxford (2011)Google Scholar
  22. 22.
    Reiner, V., Welker, V.: On the Charney–Davis and Neggers–Stanley conjectures. J. Combin. Theory Ser. A 109(2), 247–280 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schweig, J.: On the \(h\)-vector of a lattice path matroid. Electron. J. Combin. 17(1), Note 3 (2010)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Schweig, J.: Toric ideals of lattice path matroids and polymatroids. J. Pure Appl. Algebra 215(11), 2660–2665 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Simion, R.: A multi-indexed Sturm sequence of polynomials and unimodality of certain combinatorial sequences. J. Combin. Theory Ser. A 36(1), 15–22 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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
  27. 27.
    Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1(1), 9–23 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Stembridge, J.R.: Enriched \(P\)-partitions. Trans. Am. Math. Soc. 349(2), 763–788 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Stembridge, J.R.: Counterexamples to the poset conjectures of Neggers, Stanley, and Stembridge. Trans. Am. Math. Soc. 359(3), 1115–1128 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wagner, D.G.: Total positivity of Hadamard products. J. Math. Anal. Appl. 163(2), 459–483 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Welsh, D.J.A.: Matroid Theory. L. M. S. Monographs, vol. 8. Academic Press, London (1976)Google Scholar

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Authors and Affiliations

  1. 1.Laboratoire d’Informatique Fondamentale, Faculté des Sciences de LuminyAix-Marseille Université and CNRSMarseille Cedex 9France
  2. 2.Department of Computer Science, Faculty of Natural SciencesBen-Gurion University of the NegevBeer ShevaIsrael
  3. 3.Institut Montpelliérain Alexander GrothendieckUniversité de MontpellierMontpellier CedexFrance

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