ℎ*-polynomials of zonotopes

Beck, Matthias; Jochemko, Katharina; McCullough, Emily

doi:10.1090/tran/7384

$h^\ast$-polynomials of zonotopes
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Abstract:

The Ehrhart polynomial of a lattice polytope $P$ encodes information about the number of integer lattice points in positive integral dilates of $P$. The $h^\ast$-polynomial of $P$ is the numerator polynomial of the generating function of its Ehrhart polynomial. A zonotope is any projection of a higher dimensional cube. We give a combinatorial description of the $h^\ast$-polynomial of a lattice zonotope in terms of refined descent statistics of permutations and prove that the $h^\ast$-polynomial of every lattice zonotope has only real roots and therefore unimodal coefficients. Furthermore, we present a closed formula for the $h^\ast$-polynomial of a zonotope in matroidal terms which is analogous to a result by Stanley (1991) on the Ehrhart polynomial. Our results hold not only for $h^\ast$-polynomials but carry over to general combinatorial positive valuations. Moreover, we give a complete description of the convex hull of all $h^\ast$-polynomials of zonotopes in a given dimension: it is a simplicial cone spanned by refined Eulerian polynomials.

References

Matthias Beck and Benjamin Braun, Euler-Mahonian statistics via polyhedral geometry, Adv. Math. 244 (2013), 925–954. MR 3077893, DOI 10.1016/j.aim.2013.06.002
Matthias Beck, Sinai Robins, and Steven V. Sam, Corrigendum to: Positivity theorems for solid-angle polynomials [ 2663951], Beitr. Algebra Geom. 56 (2015), no. 2, 775–776. MR 3391205, DOI 10.1007/s13366-015-0259-1
Matthias Beck and Raman Sanyal, Combinatorial Reciprocity Theorems, AMS Graduate Studies in Mathematics, to appear; preprint available at http://math.sfsu.edu/ beck/crt.html.
U. Betke and P. Gritzmann, An application of valuation theory to two problems in discrete geometry, Discrete Math. 58 (1986), no. 1, 81–85. MR 820840, DOI 10.1016/0012-365X(86)90188-3
Petter Brändén, On linear transformations preserving the Pólya frequency property, Trans. Amer. Math. Soc. 358 (2006), no. 8, 3697–3716. MR 2218995, DOI 10.1090/S0002-9947-06-03856-6
Benjamin Braun, Unimodality problems in Ehrhart theory, Recent trends in combinatorics, IMA Vol. Math. Appl., vol. 159, Springer, [Cham], 2016, pp. 687–711. MR 3526428, DOI 10.1007/978-3-319-24298-9_{2}7
Francesco Brenti, $q$-Eulerian polynomials arising from Coxeter groups, European J. Combin. 15 (1994), no. 5, 417–441. MR 1292954, DOI 10.1006/eujc.1994.1046
Francesco Brenti and Volkmar Welker, $f$-vectors of barycentric subdivisions, Math. Z. 259 (2008), no. 4, 849–865. MR 2403744, DOI 10.1007/s00209-007-0251-z
R. Ehrenborg, M. Readdy, and E. Steingrímsson, Mixed volumes and slices of the cube, J. Combin. Theory Ser. A 81 (1998), no. 1, 121–126. MR 1492874, DOI 10.1006/jcta.1997.2832
Eugène Ehrhart, Sur les polyèdres rationnels homothétiques à $n$ dimensions, C. R. Acad. Sci. Paris 254 (1962), 616–618 (French). MR 130860
Takayuki Hibi, Dual polytopes of rational convex polytopes, Combinatorica 12 (1992), no. 2, 237–240. MR 1179260, DOI 10.1007/BF01204726
Katharina Jochemko, On the combinatorics of valuations, Ph.D. thesis, Freie Universität Berlin, Germany, 2015, https://refubium.fu-berlin.de/.
Katharina Jochemko and Raman Sanyal, Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 9, 2181–2208. MR 3836844, DOI 10.4171/JEMS/809
Matthias Köppe and Sven Verdoolaege, Computing parametric rational generating functions with a primal Barvinok algorithm, Electron. J. Combin. 15 (2008), no. 1, Research Paper 16, 19. MR 2383436
Martina Kubitzke and Eran Nevo, The Lefschetz property for barycentric subdivisions of shellable complexes, Trans. Amer. Math. Soc. 361 (2009), no. 11, 6151–6163. MR 2529927, DOI 10.1090/S0002-9947-09-04794-1
Jeffrey C. Lagarias and Günter M. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice, Canad. J. Math. 43 (1991), no. 5, 1022–1035. MR 1138580, DOI 10.4153/CJM-1991-058-4
Emily McCullough, On $\delta$-polynomials for lattice parallelepipeds, Master’s thesis, San Francisco State University, 2016, http://math.sfsu.edu/beck/teach/masters/emily.pdf.
P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes, Proc. London Math. Soc. (3) 35 (1977), no. 1, 113–135. MR 448239, DOI 10.1112/plms/s3-35.1.113
Peter McMullen, Valuations on lattice polytopes, Adv. Math. 220 (2009), no. 1, 303–323. MR 2462842, DOI 10.1016/j.aim.2008.09.004
James Oxley, Matroid theory, 2nd ed., Oxford Graduate Texts in Mathematics, vol. 21, Oxford University Press, Oxford, 2011. MR 2849819, DOI 10.1093/acprof:oso/9780198566946.001.0001
Thomas W. Pensyl and Carla D. Savage, Lecture hall partitions and the wreath products $C_k\wr S_n$ [reprint of MR3055684], Combinatorial number theory, De Gruyter Proc. Math., De Gruyter, Berlin, 2013, pp. 137–154. MR 3220920
Carla D. Savage and Mirkó Visontai, The $\mathbf {s}$-Eulerian polynomials have only real roots, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1441–1466. MR 3280050, DOI 10.1090/S0002-9947-2014-06256-9
Jan Schepers and Leen Van Langenhoven, Unimodality questions for integrally closed lattice polytopes, Ann. Comb. 17 (2013), no. 3, 571–589. MR 3090179, DOI 10.1007/s00026-013-0185-6
G. C. Shephard, Combinatorial properties of associated zonotopes, Canadian J. Math. 26 (1974), 302–321. MR 362054, DOI 10.4153/CJM-1974-032-5
Richard P. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math. 6 (1980), 333–342. MR 593545
Richard P. Stanley, Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Combin. Theory Ser. A 31 (1981), no. 1, 56–65. MR 626441, DOI 10.1016/0097-3165(81)90053-4
Richard P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986) Ann. New York Acad. Sci., vol. 576, New York Acad. Sci., New York, 1989, pp. 500–535. MR 1110850, DOI 10.1111/j.1749-6632.1989.tb16434.x
Richard P. Stanley, A zonotope associated with graphical degree sequences, Applied geometry and discrete mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 555–570. MR 1116376
A. Stapledon, Inequalities and Ehrhart $\delta$-vectors, Trans. Amer. Math. Soc. 361 (2009), no. 10, 5615–5626. MR 2515826, DOI 10.1090/S0002-9947-09-04776-X