Annals of Combinatorics

, Volume 23, Issue 2, pp 347–365 | Cite as

On the Relationship Between Ehrhart Unimodality and Ehrhart Positivity

  • Fu LiuEmail author
  • Liam Solus


For a given lattice polytope, two fundamental problems within the field of Ehrhart theory are (1) to determine if its (Ehrhart) \(h^*\)-polynomial is unimodal and (2) to determine if its Ehrhart polynomial has only positive coefficients. The former property of a lattice polytope is known as Ehrhart unimodality and the latter property is known as Ehrhart positivity. These two properties are often simultaneously conjectured to hold for interesting families of lattice polytopes, yet they are typically studied in parallel. As to answer a question posed at the 2017 Introductory Workshop to the MSRI Semester on Geometric and Topological Combinatorics, the purpose of this note is to show that there is no general implication between these two properties in any dimension greater than two. To do so, we investigate these two properties for families of well-studied lattice polytopes, assessing one property where previously only the other had been considered. Consequently, new examples of each phenomena are developed, some of which provide an answer to an open problem in the literature. The well-studied families of lattice polytopes considered include zonotopes, matroid polytopes, simplices of weighted projective spaces, empty lattice simplices, smooth polytopes, and \({\varvec{s}}\)-lecture hall simplices.


Ehrhart positivity Ehrhart unimodality Real-rooted polynomial Lattice polytope Weighted projective space \({\varvec{s}}\)-lecture hall simplex Empty simplex Smooth polytope 

Mathematics Subject Classification

52B20 05A20 05A15 



Fu Liu was partially supported by a Grant from the Simons Foundation #426756. She was also supported by the National Science Foundation under Grant no. DMS-1440140 while she was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. Liam Solus was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship (DMS-1606407). The authors would like to thank Alexander Postnikov for posing the motivating question of this note. This research began at the 2017 MSRI Introductory Workshop to the Semester on Geometric and Topological Combinatorics and was continued at the 2017 Mini-Workshop on Lattice Polytopes at Oberwolfach. The authors are grateful to both institutions as well as the organizers of each event.


  1. 1.
    Batyrev, V.V.: Lattice polytopes with a given \(h^\ast \)-polynomial. In: Athanasiadis, C.A., Batyrev, V.V., Dais, D.I., Henk, M., Santos, F. (eds.) Algebraic and Geometric Combinatorics. Contemp. Math., Vol. 423, pp. 1–10. Amer. Math. Soc., Providence, RI (2006)Google Scholar
  2. 2.
    Beck, M., Jochemko, K., McCullough, E.: \(h^\ast \)-polynomials of zonotopes. Trans. Amer. Math. Soc. 371(3), 2021–2042 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beck, M., Robins, S.: Computing the Continuous Discretely. Springer, New York (2007)zbMATHGoogle Scholar
  4. 4.
    Braun, B.: Unimodality problems in Ehrhart theory. In: Beveridge, A., Griggs, J.R., Hogben, L., Musiker, G., Tetali P. (eds.) Recent Trends in Combinatorics. IMA Vol. Math. Appl., Vol. 159, pp. 687–711. Springer, Cham (2016)Google Scholar
  5. 5.
    Braun, B., Davis, R.: Ehrhart series, unimodality, and integrally closed reflexive polytopes. Ann. Comb. 20(4), 705–717 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Braun, B., Davis, R., Solus, L.: Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices. Adv. in Appl. Math. 100, 122–142 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brändén, P.: Unimodality, log-concavity, real-rootedness and beyond. In: Bóna, M. (ed.) Handbook of Enumerative Combinatorics. pp. 437–483. CRC Press, Boca Raton, FL (2015)CrossRefGoogle Scholar
  8. 8.
    Brenti, F.: Unimodal, log-concave and Polya frequency sequences in combinatorics. Mem. Amer. Math. Soc. 81(413), 1–106 (1989)zbMATHGoogle Scholar
  9. 9.
    Brenti, F.: \(q\)-Eulerian polynomials arising from Coxeter groups. European J. Combin. 15(5), 417–441 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Castillo, F., Liu, F.: Berline-Vergne valuation and generalized permutohedra. Discrete Comput. Geom. 60(4), 885–908 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Castillo, F., Liu, F.: Ehrhart positivity for generalized permutohedra. In: Proceedings of FPSAC 2015. pp. 865–876. Discrete Math Theor. Comput. Sci., Nancy (2015)Google Scholar
  12. 12.
    Castillo, F., Liu, F., Nill, B., Paffenholz, A.: Smooth polytopes with negative Ehrhart coefficients. J. Combin. Theory Ser. A 160, 316–331 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    Ehrhart, E.: Sur les polyhèdres rationnels homothètiques à \(n\) dimensions. C. R. Acad. Sci. Paris 254, 616–618 (1962)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes. In: Kalai, G., Ziegler, G.M. (eds.) Polytopes-Combinatorics and Computation. pp. 43–73. Birkhäuser, Basel (2000)CrossRefGoogle Scholar
  16. 16.
    Hibi, T.: Algebraic Combinatorics on Convex Polytopes. Carslaw publications, Glebe (1992)zbMATHGoogle Scholar
  17. 17.
    Hibi, T., Higashitani, A., Tsuchiya, A., Yoshida, K.: Ehrhart polynomials with negative coefficients. Graphs Combin. 35(1), 363–371 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Knauer, K., Martínez-Sandoval, L., Ramírez Alfonsín, J.L.: On lattice path matroid polytopes: integer points and Ehrhart polynomial. Discrete Comput. Geom. 60(3), 698–719 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Köppe, M., Verdoolaege, S.: Computing parametric rational generating functions with a primal Barvinok algorithm. Electron. J. Combin. 15(1), #P16 (2008)Google Scholar
  20. 20.
    Liu, F.: Ehrhart polynomials of cyclic polytopes. J. Combin. Theory Ser. A 111(1), 111–127 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liu, F.: On positivity of Ehrhart polynomials. arXiv:1711.09962 (2017)
  22. 22.
    Mini-Workshop: Lattice polytopes: methods, advances, applications. Abstracts from the mini-workshop held September 17–23, 2017. Organized by Hibi, T., Higashitani, A., Jochemko, K., Nill, B.. Oberwolfach Rep. 14(3), 2659–2701 (2017)Google Scholar
  23. 23.
    Payne, S.: Ehrhart series and lattice triangulations. Discrete Comput. Geom. 40(3), 365–376 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rodriguez-Villegas, F.: On the zeros of certain polynomials. Proc. Amer. Math. Soc. 130(8), 2251–2254 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Savage, C.D.: The mathematics of lecture hall partitions. J. Combin. Theory Ser. A 144, 443–475 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Savage, C.D., Schuster, M.J.: Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences. J. Combin. Theory Ser. A 119(4), 850–870 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Savage, C.D., Visontai, M.: The \(s\)-Eulerian polynomials have only real roots. Trans. Amer. Math. Soc. 367(2), 1441–1466 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Solus, L.: Simplices for numeral systems. Trans. Amer. Math. Soc. 371(3), 2089–2107 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Stanley, R.P.: Enumerative Combinatorics. Vol. 1. Cambridge Studies in Advanced Mathematics, Vol. 49. Cambridge University Press, Cambridge (1997)Google Scholar
  31. 31.
    Stanley, R.P.: Positivity of Ehrhart polynomial coefficients (answer). MathOverflow. (2015)
  32. 32.
    Stanley, R.P.: Two enumerative results on cycles of permutations. European J. Combin. 32(6), 937–943 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, Vol. 152. Springer-Verlag, New York (1995)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA
  2. 2.Matematik, KTHStockholmSweden

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