{
  "id": "1702.06666",
  "version": "v1",
  "published": "2017-02-22T03:44:09.000Z",
  "updated": "2017-02-22T03:44:09.000Z",
  "title": "Gamma-positivity of variations of Eulerian polynomials",
  "authors": [
    "John Shareshian",
    "Michelle L. Wachs"
  ],
  "comment": "29 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "An identity of Chung, Graham and Knuth involving binomial coefficients and Eulerian numbers motivates our study of a class of polynomials that we call binomial-Eulerian polynomials. These polynomials share several properties with the Eulerian polynomials. For one thing, they are $h$-polynomials of simplicial polytopes, which gives a geometric interpretation of the fact that they are palindromic and unimodal. A formula of Foata and Sch\\\"utzenberger shows that the Eulerian polynomials have a stronger property, namely $\\gamma$-positivity, and a formula of Postnikov, Reiner and Williams does the same for the binomial-Eulerian polynomials. We obtain $q$-analogs of both the Foata-Sch\\\"utzenberger formula and an alternative to the Postnikov-Reiner-Williams formula, and we show that these $q$-analogs are specializations of analogous symmetric function identities. Algebro-geometric interpretations of these symmetric function analogs are presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-22T03:44:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05E05",
      "05E10",
      "05E45",
      "52B05"
    ],
    "keywords": [
      "gamma-positivity",
      "variations",
      "binomial-eulerian polynomials",
      "analogous symmetric function identities",
      "symmetric function analogs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}