{
  "id": "1609.08596",
  "version": "v1",
  "published": "2016-09-27T19:45:29.000Z",
  "updated": "2016-09-27T19:45:29.000Z",
  "title": "$h^\\ast$-polynomials of zonotopes",
  "authors": [
    "Matthias Beck",
    "Katharina Jochemko",
    "Emily McCullough"
  ],
  "comment": "20 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-27T19:45:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B12",
      "52B20",
      "52B45",
      "26C10",
      "05A10",
      "05A15"
    ],
    "keywords": [
      "ehrhart polynomial",
      "lattice zonotope",
      "integer lattice points",
      "general combinatorial positive valuations",
      "higher dimensional cube"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}