{
  "id": "2108.11016",
  "version": "v1",
  "published": "2021-08-25T02:25:55.000Z",
  "updated": "2021-08-25T02:25:55.000Z",
  "title": "On the Number of 2-Hooks and 3-Hooks of Integer Partitions",
  "authors": [
    "Eleanor Mcspirit",
    "Kristen Scheckelhoff"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $p_t(a,b;n)$ denote the number of partitions of $n$ such that the number of $t$ hooks is congruent to $a \\bmod{b}$. For $t\\in \\{2, 3\\}$, arithmetic progressions $r_1 \\bmod{m_1}$ and $r_2 \\bmod{m_2}$ on which $p_t(r_1,m_1; m_2 n + r_2)$ vanishes were established in recent work by Bringmann, Craig, Males, and Ono using the theory of modular forms. Here we offer a direct combinatorial proof of this result using abaci and the theory of $t$-cores and $t$-quotients.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-25T02:25:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integer partitions",
      "direct combinatorial proof",
      "arithmetic progressions",
      "modular forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}