{
  "id": "1212.3505",
  "version": "v1",
  "published": "2012-12-14T15:45:48.000Z",
  "updated": "2012-12-14T15:45:48.000Z",
  "title": "On the Maximum Number of k-Hooks of Partitions of n",
  "authors": [
    "Anna R. B. Fan",
    "Harold R. L. Yang",
    "Rebecca T. Yu"
  ],
  "comment": "14 pages, 8 figures",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $\\alpha_k(\\lambda)$ denote the number of $k$-hooks in a partition $\\lambda$ and let $b(n,k)$ be the maximum value of $\\alpha_k(\\lambda)$ among partitions of $n$. Amdeberhan posed a conjecture on the generating function of $b(n,1)$. We give a proof of this conjecture. In general, we obtain a formula that can be used to determine $b(n,k)$. This leads to a generating function formula for $b(n,k)$. We introduce the notion of nearly $k$-triangular partitions. We show that for any $n$, there is a nearly $k$-triangular partition which can be transformed into a partition of $n$ that attains the maximum number of $k$-hooks. The operations for the transformation enable us to compute the number $b(n,k)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-14T15:45:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A17"
    ],
    "keywords": [
      "maximum number",
      "triangular partition",
      "maximum value",
      "conjecture",
      "generating function formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.3505F"
    }
  }
}