{
  "id": "2201.06630",
  "version": "v2",
  "published": "2022-01-17T20:57:45.000Z",
  "updated": "2022-05-19T13:54:31.000Z",
  "title": "Distributions of Hook lengths in integer partitions",
  "authors": [
    "Michael Griffin",
    "Ken Ono",
    "Wei-Lun Tsai"
  ],
  "comment": "Dedication added: \"In memory of Christine Bessenrodt\" Minor edits to previous version",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of $t$-hooks in the partitions of $n$. We prove that the limiting distribution is normal with mean $\\mu_t(n)\\sim \\frac{\\sqrt{6n}}{\\pi}-\\frac{t}{2}$ and variance $\\sigma_t^2(n)\\sim \\frac{(\\pi^2-6)\\sqrt{6n}}{2\\pi^3}.$ Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed $t\\geq 4$ in partitions of $n$ converge to a shifted Gamma distribution with parameter $k=(t-1)/2$ and scale $\\theta=\\sqrt{2/(t-1)}.$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-05-19T13:54:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P82",
      "05A17"
    ],
    "keywords": [
      "integer partitions",
      "hook lengths play",
      "shifted gamma distribution",
      "limiting distribution",
      "mathematics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}