{
  "id": "1303.2302",
  "version": "v2",
  "published": "2013-03-10T09:51:28.000Z",
  "updated": "2013-03-15T10:04:21.000Z",
  "title": "A symmetric unimodal decomposition of the derangement polynomial of type $B$",
  "authors": [
    "Christos A. Athanasiadis",
    "Christina Savvidou"
  ],
  "comment": "Changes in Remark 7.3 and the bibliography",
  "categories": [
    "math.CO"
  ],
  "abstract": "The derangement polynomial $d_n (x)$ for the symmetric group enumerates derangements by the number of excedances. The derangement polynomial $d^B_n(x)$ for the hyperoctahedral group is a natural type $B$ analogue. A new combinatorial formula for this polynomial is given in this paper. This formula implies that $d^B_n (x)$ decomposes as a sum of two nonnegative, symmetric and unimodal polynomials whose centers of symmetry differ by a half and thus provides a new transparent proof of its unimodality. A geometric interpretation, analogous to Stanley's interpretation of $d_n (x)$ as the local $h$-polynomial of the barycentric subdivision of the simplex, is given to one of the summands of this decomposition. This interpretation leads to a unimodal decomposition and a new formula for the Eulerian polynomial of type $B$. The various decomposing polynomials introduced here are also studied in terms of recurrences, generating functions, combinatorial interpretations, expansions and real-rootedness.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-03-15T10:04:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15",
      "05E45"
    ],
    "keywords": [
      "derangement polynomial",
      "symmetric unimodal decomposition",
      "symmetric group enumerates derangements",
      "interpretation",
      "natural type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.2302A"
    }
  }
}