{
  "id": "2109.09370",
  "version": "v1",
  "published": "2021-09-20T08:44:05.000Z",
  "updated": "2021-09-20T08:44:05.000Z",
  "title": "Clustering of consecutive numbers in permutations avoiding a pattern and in separable permutations",
  "authors": [
    "Ross G. Pinsky"
  ],
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "Let $S_n$ denote the set of permutations of $[n]:=\\{1,\\cdots, n\\}$, and denote a permutation $\\sigma\\in S_n$ by $\\sigma=\\sigma_1\\sigma_2\\cdots \\sigma_n$. For $l\\ge2$ an integer, let $A^{(n)}_{l;k}\\subset S_n$ denote the event that the set of $l$ consecutive numbers $\\{k, k+1,\\cdots, k+l-1\\}$ appears in a set of consecutive positions: $\\{k,k+1,\\cdots, k+l-1\\}=\\{\\sigma_a,\\sigma_{a+1},\\cdots, \\sigma_{a+l-1}\\}$, for some $a$. For $\\tau\\in S_m$, let $S_n(\\tau)$ denote the set of $\\tau$-avoiding permutations in $S_n$, and let $P_n^{\\text{av}(\\tau)}$ denote the uniform probability measure on $S_n(\\tau)$. Also, let $S_n^{\\text{sep}}$ denote the set of separable permutations in $S_n$, and let $P_n^{\\text{sep}}$ denote the uniform probability measure on $S_n^{\\text{sep}}$. We investigate the quantities $P_n^{\\text{av}(\\tau)}(A^{(n)}_{l;k})$ and $P_n^{\\text{sep}}(A^{(n)}_{l;k})$ for fixed $n$, and the limiting behavior as $n\\to\\infty$. We also consider the asymptotic properties of this limiting behavior as $l\\to\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-20T08:44:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "60C05"
    ],
    "keywords": [
      "consecutive numbers",
      "separable permutations",
      "permutations avoiding",
      "uniform probability measure",
      "limiting behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}