{
  "id": "1212.6468",
  "version": "v1",
  "published": "2012-12-28T06:10:54.000Z",
  "updated": "2012-12-28T06:10:54.000Z",
  "title": "Decomposition of Triply Rooted Trees",
  "authors": [
    "William Y. C. Chen",
    "Janet F. F. Peng",
    "Harold R. L. Yang"
  ],
  "comment": "10 pages, 5 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we give a decomposition of triply rooted trees into three doubly rooted trees. This leads to a combinatorial interpretation of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory, and proved by Younsi by using the Hurwitz identity on multivariate Abel polynomials. We also give a bijection between the set of functions from $[n+1]$ to $[n]$ and the set of triply rooted trees on $[n]$, which leads to the refined enumeration of functions from $[n+1]$ to $[n]$ with respect to the number of elements in the orbit of $n+1$ and the number of periodic points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-28T06:10:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A19"
    ],
    "keywords": [
      "triply rooted trees",
      "decomposition",
      "pac-bayesian machine learning theory",
      "multivariate abel polynomials",
      "periodic points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.6468C"
    }
  }
}