{
  "id": "2203.12510",
  "version": "v1",
  "published": "2022-03-23T16:12:56.000Z",
  "updated": "2022-03-23T16:12:56.000Z",
  "title": "Exact formula and asymptotic behavior for the expected number of inversions in a random permutation avoiding a pattern of length three",
  "authors": [
    "Ross G. Pinsky"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "For $\\tau\\in S_3$, let $S_n(\\tau)$ denote the set of permutations in $S_n$ which avoid the pattern $\\tau$, and let $E_n^\\tau$ denote the expectation with respect to the uniformly random probability measure on $S_n(\\tau)$. Let $\\mathcal{I}_n(\\sigma)$ denote the number of inversions in $\\sigma\\in S_n$. We study $E_n^\\tau\\mathcal{I}_n$ for $\\tau\\in\\{231,132,213,312\\}\\subset S_3$. We prove that $$ E_n^{231}\\mathcal{I}_n=E_n^{312}\\mathcal{I}_n=\\frac12\\frac{n!(n+1)!4^n}{(2n)!}-\\frac12(3n+1), $$ and that $$ E_n^{132}\\mathcal{I}_n=E_n^{213}\\mathcal{I}_n=\\frac12(n-1)n-E_n^{231}\\mathcal{I}_n. $$ From the first equation it follows that $$ E_n^{231}\\mathcal{I}_n=E_n^{312}\\mathcal{I}_n\\sim\\frac{\\sqrt\\pi}2n^\\frac32. $$ We also show that the variance $\\text{Var}_{P_n^{\\tau}}(\\mathcal{I}_n)$ of $\\mathcal{I}_n$ under $P_n^\\tau$ satisfies $$ \\text{Var}_{P_n^{\\tau}}(\\mathcal{I}_n)\\sim (\\frac56-\\frac\\pi4)n^3\\approx 0.048n^3,\\ \\text{for}\\ \\tau\\in\\{231,132,213,312\\}. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-23T16:12:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "05A05"
    ],
    "keywords": [
      "random permutation avoiding",
      "exact formula",
      "asymptotic behavior",
      "expected number",
      "inversions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}