{
  "id": "0907.5351",
  "version": "v1",
  "published": "2009-07-30T14:14:47.000Z",
  "updated": "2009-07-30T14:14:47.000Z",
  "title": "Profiles of permutations",
  "authors": [
    "Michael Lugo"
  ],
  "comment": "23 pages, 1 figure. Submitted to Electronic Journal of Combinatorics",
  "journal": "Electronic Journal of Combinatorics, vol. 16(1) (2009), #R99",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $\\sigma$ and, on the other hand, permutations selected according to the Ewens distribution with parameter $\\sigma$. In particular we show that the asymptotic expected number of cycles of random permutations of $[n]$ with all cycles even, with all cycles odd, and chosen from the Ewens distribution with parameter 1/2 are all ${1 \\over 2} \\log n + O(1)$, and the variance is of the same order. Furthermore, we show that in permutations of $[n]$ chosen from the Ewens distribution with parameter $\\sigma$, the probability of a random element being in a cycle longer than $\\gamma n$ approaches $(1-\\gamma)^\\sigma$ for large $n$. The same limit law holds for permutations with cycles carrying multiplicative weights with average $\\sigma$. We draw parallels between the Ewens distribution and the asymptotic-density case and explain why these parallels should exist using permutations drawn from weighted Boltzmann distributions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-07-30T14:14:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A16",
      "60C05"
    ],
    "keywords": [
      "ewens distribution",
      "random permutations",
      "limit law holds",
      "cycle lengths",
      "draw parallels"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.5351L"
    }
  }
}