{
  "id": "2311.15013",
  "version": "v1",
  "published": "2023-11-25T12:34:45.000Z",
  "updated": "2023-11-25T12:34:45.000Z",
  "title": "Inequalities and asymptotics for hook numbers in restricted partitions",
  "authors": [
    "William Craig",
    "Madeline Locus Dawsey",
    "Guo-Niu Han"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "In this paper, we consider the asymptotic properties of hook numbers of partitions in restricted classes. More specifically, we compare the frequency with which partitions into odd parts and partitions into distinct parts have hook numbers equal to $h \\geq 1$ by deriving an asymptotic formula for the total number of hooks equal to $h$ that appear among partitions into odd and distinct parts, respectively. We use these asymptotic formulas to prove a recent conjecture of the first author and collaborators that for $h \\geq 2$ and $n \\gg 0$, partitions into odd parts have, on average, more hooks equal to $h$ than do partitions into distinct parts. We also use our asymptotics to prove certain probabilistic statements about how hooks distribute in the rows of partitions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-25T12:34:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "restricted partitions",
      "distinct parts",
      "asymptotic formula",
      "hooks equal",
      "odd parts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}