On the relationship between Ehrhart unimodality and Ehrhart positivity. (English) Zbl 1418.52004

Summary: 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 \(\boldsymbol{s}\)-lecture hall simplices.


52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
05A20 Combinatorial inequalities
05A15 Exact enumeration problems, generating functions
Full Text: DOI arXiv


[1] Batyrev, V.V.: Lattice polytopes with a given \[h^\ast\] h*-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) · Zbl 1121.52026
[2] Beck, M., Jochemko, K., McCullough, E.: \[h^\ast\] h*-polynomials of zonotopes. Trans. Amer. Math. Soc. 371(3), 2021-2042 (2019) · Zbl 1402.05100 · doi:10.1090/tran/7384
[3] Beck, M., Robins, S.: Computing the Continuous Discretely. Springer, New York (2007) · Zbl 1114.52013
[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) · Zbl 1366.52014
[5] Braun, B., Davis, R.: Ehrhart series, unimodality, and integrally closed reflexive polytopes. Ann. Comb. 20(4), 705-717 (2016) · Zbl 1379.52013 · doi:10.1007/s00026-016-0337-6
[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) · Zbl 1396.52020 · doi:10.1016/j.aam.2018.06.003
[7] Brändén, P.; Bóna, M. (ed.), Unimodality, log-concavity, real-rootedness and beyond, 437-483 (2015), Boca Raton, FL · Zbl 1327.05051 · doi:10.1201/b18255-10
[8] Brenti, F.: Unimodal, log-concave and Polya frequency sequences in combinatorics. Mem. Amer. Math. Soc. 81(413), 1-106 (1989) · Zbl 0697.05011
[9] Brenti, \[F.: q\] q-Eulerian polynomials arising from Coxeter groups. European J. Combin. 15(5), 417-441 (1994) · Zbl 0809.05012 · doi:10.1006/eujc.1994.1046
[10] Castillo, F., Liu, F.: Berline-Vergne valuation and generalized permutohedra. Discrete Comput. Geom. 60(4), 885-908 (2018) · Zbl 1401.52024 · doi:10.1007/s00454-017-9950-3
[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) · Zbl 1362.05130
[12] Castillo, F., Liu, F., Nill, B., Paffenholz, A.: Smooth polytopes with negative Ehrhart coefficients. J. Combin. Theory Ser. A 160, 316-331 (2018) · Zbl 1402.52017 · doi:10.1016/j.jcta.2018.06.014
[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) · Zbl 1207.52015 · doi:10.1007/s00454-008-9080-z
[14] Ehrhart, E.: Sur les polyhèdres rationnels homothètiques à \[n\] n dimensions. C. R. Acad. Sci. Paris 254, 616-618 (1962) · Zbl 0100.27601
[15] Gawrilow, E.; Joswig, M.; Kalai, G. (ed.); Ziegler, GM (ed.), Polymake: a framework for analyzing convex polytopes, 43-73 (2000), Basel · Zbl 0960.68182 · doi:10.1007/978-3-0348-8438-9_2
[16] Hibi, T.: Algebraic Combinatorics on Convex Polytopes. Carslaw publications, Glebe (1992) · Zbl 0772.52008
[17] Hibi, T., Higashitani, A., Tsuchiya, A., Yoshida, K.: Ehrhart polynomials with negative coefficients. Graphs Combin. 35(1), 363-371 (2019) · Zbl 1409.52012 · doi:10.1007/s00373-018-1990-9
[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) · Zbl 1494.52014 · doi:10.1007/s00454-018-9965-4
[19] Köppe, M., Verdoolaege, S.: Computing parametric rational generating functions with a primal Barvinok algorithm. Electron. J. Combin. 15(1), #P16 (2008) · Zbl 1180.52014
[20] Liu, F.: Ehrhart polynomials of cyclic polytopes. J. Combin. Theory Ser. A 111(1), 111-127 (2005) · Zbl 1066.05015 · doi:10.1016/j.jcta.2004.11.011
[21] Liu, F.: On positivity of Ehrhart polynomials. arXiv:1711.09962 (2017) · Zbl 1435.52007
[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) · Zbl 1394.00017
[23] Payne, S.: Ehrhart series and lattice triangulations. Discrete Comput. Geom. 40(3), 365-376 (2008) · Zbl 1159.52017 · doi:10.1007/s00454-007-9002-5
[24] Rodriguez-Villegas, F.: On the zeros of certain polynomials. Proc. Amer. Math. Soc. 130(8), 2251-2254 (2002) · Zbl 0992.12001 · doi:10.1090/S0002-9939-02-06454-7
[25] Savage, C.D.: The mathematics of lecture hall partitions. J. Combin. Theory Ser. A 144, 443-475 (2016) · Zbl 1343.05032 · doi:10.1016/j.jcta.2016.06.006
[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) · Zbl 1237.05017 · doi:10.1016/j.jcta.2011.12.005
[27] Savage, C.D., Visontai, M.: The \[s\] s-Eulerian polynomials have only real roots. Trans. Amer. Math. Soc. 367(2), 1441-1466 (2015) · Zbl 1316.05002 · doi:10.1090/S0002-9947-2014-06256-9
[28] Solus, L.: Simplices for numeral systems. Trans. Amer. Math. Soc. 371(3), 2089-2107 (2019) · Zbl 1406.52024 · doi:10.1090/tran/7424
[29] Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333-342 (1980) · Zbl 0812.52012 · doi:10.1016/S0167-5060(08)70717-9
[30] Stanley, R.P.: Enumerative Combinatorics. Vol. 1. Cambridge Studies in Advanced Mathematics, Vol. 49. Cambridge University Press, Cambridge (1997) · Zbl 0889.05001
[31] Stanley, R.P.: Positivity of Ehrhart polynomial coefficients (answer). MathOverflow. https://mathoverflow.net/questions/185723/positivity-of-ehrhart-polynomial-coefficients (2015)
[32] Stanley, R.P.: Two enumerative results on cycles of permutations. European J. Combin. 32(6), 937-943 (2011) · Zbl 1238.05015 · doi:10.1016/j.ejc.2011.01.011
[33] Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, Vol. 152. Springer-Verlag, New York (1995) · Zbl 0823.52002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.