{
  "id": "1906.07169",
  "version": "v1",
  "published": "2019-06-16T12:40:07.000Z",
  "updated": "2019-06-16T12:40:07.000Z",
  "title": "The limiting distribution of the hook length of a randomly chosen cell in a random Young diagram",
  "authors": [
    "Ljuben Mutafchiev"
  ],
  "comment": "14 pages. arXiv admin note: text overlap with arXiv:1306.6155, arXiv:1407.3639",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $p(n)$ be the number of all integer partitions of the positive integer $n$ and let $\\lambda$ be a partition, selected uniformly at random from among all such $p(n)$ partitions. It is known that each partition $\\lambda$ has a unique graphical representation, composed by $n$ non-overlapping cells in the plane called Young diagram. As a second step of our sampling experiment, we select a cell $c$ uniformly at random from among all $n$ cells of the Young diagram of the partition $\\lambda$. For large $n$, we study the asymptotic behavior of the hook length $Z_n=Z_n(\\lambda,c)$ of the cell $c$ of a random partituion $\\lambda$. This two-step sampling procedure suggests a product probability measure, which assigns the probability $1/np(n)$ to each pair $(\\lambda,c)$. With respect to this probability measure, we show that the random variable $\\pi Z_n/\\sqrt{6n}$ converges weakly, as $n\\to\\infty$, to a random variable whose probability density function equals $6y/\\pi^2 (e^y-1)$ if $0<y<\\infty$, and zero elsewhere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-16T12:40:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P82",
      "05A17",
      "60F05",
      "60C05"
    ],
    "keywords": [
      "random young diagram",
      "randomly chosen cell",
      "hook length",
      "limiting distribution",
      "probability density function equals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}