{
  "id": "1502.07674",
  "version": "v1",
  "published": "2015-02-26T18:46:50.000Z",
  "updated": "2015-02-26T18:46:50.000Z",
  "title": "Another combinatorial proof of a result of Zagier and Stanley",
  "authors": [
    "Ricky X. F. Chen",
    "Christian M. Reidys"
  ],
  "comment": "14 pages. This is the first part from division of the paper: arXiv:1411.5552v2 [math.CO], into two parts. The other part will be uploaded later and after uploading the second part, arXiv:1411.5552v2 [math.CO] will be removed. Comments are welcome",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we present another combinatorial proof for the result of Zagier and Stanley, that the number of $n$-cycles $\\omega$, for which $\\omega(12\\cdots n)$ has exactly $k$ cycles is $0$, if $n-k$ is odd and $\\frac{2C(n+1,k)}{n(n+1)}$, otherwise, where $C(n,k)$ is the unsigned Stirling number of the first kind. To this end, we generalize permutations to plane permutations and study exceedances via a natural transposition action on plane permutations. Furthermore, based on a \"reflection principle\" argument, we give a refinement of a recurrence satisfied by the numbers counting one-face hypermaps which was recently obtained by Chapuy by counting bipartite unicellular maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-26T18:46:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15"
    ],
    "keywords": [
      "combinatorial proof",
      "plane permutations",
      "counting bipartite unicellular maps",
      "numbers counting one-face hypermaps",
      "natural transposition action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150207674C"
    }
  }
}