{
  "id": "1406.3162",
  "version": "v2",
  "published": "2014-06-12T09:25:16.000Z",
  "updated": "2014-06-24T00:05:26.000Z",
  "title": "A combinatorial interpretation of the $κ^{\\star}_{g}(n)$ coefficients",
  "authors": [
    "Thomas J. X. Li",
    "Christian M. Reidys"
  ],
  "comment": "22 pages, 5 figures. arXiv admin note: text overlap with arXiv:1202.3252 by other authors",
  "categories": [
    "math.CO"
  ],
  "abstract": "Studying the virtual Euler characteristic of the moduli space of curves, Harer and Zagier compute the generating function $C_g(z)$ of unicellular maps of genus $g$. They furthermore identify coefficients, $\\kappa^{\\star}_{g}(n)$, which fully determine the series $C_g(z)$. The main result of this paper is a combinatorial interpretation of $\\kappa^{\\star}_{g}(n)$. We show that these enumerate a class of unicellular maps, which correspond $1$-to-$2^{2g}$ to a specific type of trees, referred to as O-trees. O-trees are a variant of the C-decorated trees introduced by Chapuy, F\\'{e}ray and Fusy. We exhaustively enumerate the number $s_{g}(n)$ of shapes of genus $g$ with $n$ edges, which is a specific class of unicellular maps with vertex degree at least three. Furthermore we give combinatorial proofs for expressing the generating functions $C_g(z)$ and $S_g(z)$ for unicellular maps and shapes in terms of $\\kappa^{\\star}_{g}(n)$, respectively. We then prove a two term recursion for $\\kappa^{\\star}_{g}(n)$ and that for any fixed $g$, the sequence $\\{\\kappa_{g,t}\\}_{t=0}^g$ is log-concave, where $\\kappa^{\\star}_{g}(n)= \\kappa_{g,t}$, for $n=2g+t-1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-24T00:05:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19"
    ],
    "keywords": [
      "combinatorial interpretation",
      "unicellular maps",
      "coefficients",
      "virtual euler characteristic",
      "generating function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.3162L"
    }
  }
}