{
  "id": "1906.05191",
  "version": "v1",
  "published": "2019-06-12T15:01:46.000Z",
  "updated": "2019-06-12T15:01:46.000Z",
  "title": "On the joint distribution of cyclic valleys and excedances over conjugacy classes of $\\mathfrak{S}_{n}$",
  "authors": [
    "M. Crossan Cooper",
    "William S. Jones",
    "Yan Zhuang"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We derive a formula expressing the joint distribution of the cyclic valley number and excedance number statistics over a fixed conjugacy class of the symmetric group in terms of Eulerian polynomials. Our proof uses a slight extension of Sun and Wang's cyclic valley-hopping action as well as a formula of Brenti. Along the way, we give a new proof for the $\\gamma$-positivity of the excedance number distribution over each fixed conjugacy class along with a combinatorial interpretation of the $\\gamma$-coefficients.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-12T15:01:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A05",
      "05E18"
    ],
    "keywords": [
      "joint distribution",
      "fixed conjugacy class",
      "excedance number statistics",
      "excedance number distribution",
      "wangs cyclic valley-hopping action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}