{
  "id": "2211.12090",
  "version": "v1",
  "published": "2022-11-22T08:46:11.000Z",
  "updated": "2022-11-22T08:46:11.000Z",
  "title": "Clustering of consecutive numbers in permutations avoiding a pattern of length three",
  "authors": [
    "Ross G. Pinsky"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "For $\\eta\\in S_3$, let $S_n^{\\text{av}(\\eta)}$ denote the set of permutations in $S_n$ that avoid the pattern $\\eta$, and let $E_n^{\\text{av}(\\eta)}$ denote the expectation with respect to the uniform probability measure on $S_n^{\\text{av}(\\eta)}$. For $n\\ge k\\ge2$ and $\\tau\\in S_k^{\\text{av}(\\eta)}$, let $N_n^{k}(\\sigma)$ denote the number of occurrences of $k$ consecutive numbers appearing in $k$ consecutive positions in $\\sigma\\in S_n^{\\text{av}(\\eta)}$, and let $N_n^{k;\\tau}(\\sigma)$ denote the number of such occurrences for which the order of the appearance of the $k$ numbers is the pattern $\\tau$. We obtain explicit formula formulas for $E_n^{\\text{av}(\\eta)}N_n^{k;\\tau}$ and $E_n^{\\text{av}(\\eta)}N_n^k$, for all $2\\le k\\le n$, all $\\eta\\in S_3$ and all $\\tau\\in S_k^{\\text{av}(\\eta)}$. These exact formulas then yield asymptotic formulas as $n\\to\\infty$ with $k$ fixed, and as $n\\to\\infty$ with $k=k_n\\to\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-22T08:46:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "05A05"
    ],
    "keywords": [
      "consecutive numbers",
      "permutations avoiding",
      "uniform probability measure",
      "yield asymptotic formulas",
      "explicit formula formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}