{
  "id": "1907.13082",
  "version": "v1",
  "published": "2019-07-30T17:17:46.000Z",
  "updated": "2019-07-30T17:17:46.000Z",
  "title": "Alternatingly increasing property and bi-gamma-positivity of polynomials",
  "authors": [
    "Shi-Mei Ma",
    "Jun Ma",
    "Yeong-Nan Yeh"
  ],
  "comment": "31 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A polynomial p(z) of degree d is alternatingly increasing if and only if it can be decomposed into a sum p(z)=a(z)+zb(z), where a(z) and b(z) are symmetric and unimodal polynomials with dega(z)=d and degb(z)<d. We say that p(z) is bi-gamma-positive if a(z) and b(z) are both gamma positive. In this paper, we present a unified elementary proof of the bi-gamma-positivity of several polynomials that appear often in algebraic, topological and geometric combinatorics, including q-Eulerian polynomials of types A and B, the descent polynomials of multipermutations and signed multipermutations, Eulerian and derangement polynomials of colored permutations. As an application, we get that these polynomials are all alternatingly increasing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-30T17:17:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05E45"
    ],
    "keywords": [
      "alternatingly increasing property",
      "bi-gamma-positivity",
      "derangement polynomials",
      "unimodal polynomials",
      "geometric combinatorics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}