{
  "id": "2404.06598",
  "version": "v1",
  "published": "2024-04-09T20:00:13.000Z",
  "updated": "2024-04-09T20:00:13.000Z",
  "title": "Extremes of generalized inversions on symmetric groups",
  "authors": [
    "Philip Dörr"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The classical inversion statistic on symmetric groups is the sum of all indicators $\\textbf{1}\\{\\pi(i) > \\pi(j)\\}$ for a random permutation $\\pi = (\\pi(1), \\ldots, \\pi(n))$ and the pairs $(i,j)$ with $1 \\leq i < j \\leq n$. The descent statistic counts all $i \\in \\{1, \\ldots, n-1\\}$ with $\\pi(i) > \\pi(i+1)$. The number of inversions can be generalized by restricting the indicators to pairs $(i,j)$ with $i<j$ and $|i-j| \\leq d$ for some $d \\in \\{1, \\ldots, n-1\\}$. Likewise, the number of descents can be generalized by counting all $i \\in \\{1, \\ldots, n-d\\}$ with $\\pi(i) > \\pi(i+d)$. These generalized statistics can be further extended to the signed and even-signed permutation groups. The bandwidth index $d$ can be chosen in dependence of $n$, and the magnitude of $d$ is significant for asymptotic considerations. It is known that each of these statistics is asymptotically normal for suitable choices of $d$. In this paper we prove the bivariate asymptotic normality and determine the extreme value asymptotics of generalized inversions and descents.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-09T20:00:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G70",
      "05A16",
      "20F55"
    ],
    "keywords": [
      "symmetric groups",
      "generalized inversions",
      "descent statistic counts",
      "bivariate asymptotic normality",
      "extreme value asymptotics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}