{
  "id": "2209.13563",
  "version": "v1",
  "published": "2022-09-27T17:35:06.000Z",
  "updated": "2022-09-27T17:35:06.000Z",
  "title": "The asymptotic number of score sequences",
  "authors": [
    "Brett Kolesnik"
  ],
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "A tournament on a graph is an orientation of its edges. The score sequence lists the in-degrees in non-decreasing order. Works by Winston and Kleitman (1983) and Kim and Pittel (2000) showed that the number $S_n$ of score sequences on the complete graph $K_n$ satisfies $S_n=\\Theta(4^n/n^{5/2})$. In this note, we observe that by combining recent combinatorial developments related to score sequences with the limit theory for discrete infinitely divisible distributions it follows that $n^{5/2}S_n/4^n\\to 0.392\\ldots$, as conjectured by Tak\\'acs (1986).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-27T17:35:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A16",
      "05C20",
      "05C30",
      "11P21",
      "51M20",
      "52B05",
      "60E07"
    ],
    "keywords": [
      "asymptotic number",
      "score sequence lists",
      "complete graph",
      "combinatorial developments",
      "limit theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}