{
  "id": "1412.3685",
  "version": "v1",
  "published": "2014-12-11T15:46:11.000Z",
  "updated": "2014-12-11T15:46:11.000Z",
  "title": "Acyclic orientations and poly-Bernoulli numbers",
  "authors": [
    "P. J. Cameron",
    "C. A. Glass",
    "R. U. Schumacher"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In 1997, Masanobu Kaneko defined \\emph{poly-Bernoulli numbers}, which bear much the same relation to polylogarithms as Berunoulli numbers do to logarithms. In 2008, Chet Brewbaker described a counting problem whose solution can be identified with the poly-Bernoulli numbers with negative index, the \\emph{lonesum matrices}. The main aim of this paper is to give formulae for the number of acyclic orientations of a complete bipartite graph, or of a complete bipartite graph with one edge added or removed. Our formula shows that the number of acyclic orientations of $K_{n_1,n_2}$ is equal to the poly-Bernoulli number $B_{n_1}^{(-n_2)}$. We also give a simple bijective identification of acyclic orientations and lonesum matrices. We make some remarks on the context of our result, which are expanded in another paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-11T15:46:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C30"
    ],
    "keywords": [
      "acyclic orientations",
      "poly-bernoulli number",
      "complete bipartite graph",
      "chet brewbaker",
      "berunoulli numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.3685C"
    }
  }
}