{
  "id": "2006.06724",
  "version": "v1",
  "published": "2020-06-11T18:31:06.000Z",
  "updated": "2020-06-11T18:31:06.000Z",
  "title": "Tree/Endofunction Bijections and Concentration Inequalities",
  "authors": [
    "Steven Heilman"
  ],
  "comment": "15 pages, 3 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We demonstrate a method for proving precise concentration inequalities in uniformly random trees on $n$ vertices, where $n\\geq1$ is a fixed positive integer. The method uses a bijection between mappings $f\\colon\\{1,\\ldots,n\\}\\to\\{1,\\ldots,n\\}$ and doubly rooted trees on $n$ vertices. The main application is a concentration inequality for the number of vertices connected to an independent set in a uniformly random tree, which is then used to prove partial unimodality of its independent set sequence. So, we give probabilistic arguments for inequalities that often use combinatorial arguments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-11T18:31:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "concentration inequality",
      "tree/endofunction bijections",
      "uniformly random tree",
      "independent set sequence",
      "proving precise concentration inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}