{
  "id": "1506.06186",
  "version": "v1",
  "published": "2015-06-20T00:08:16.000Z",
  "updated": "2015-06-20T00:08:16.000Z",
  "title": "A combinatorial proof of a relationship between maximal $(2k-1,2k+1)$ and $(2k-1,2k,2k+1)$-cores",
  "authors": [
    "Rishi Nath",
    "James A. Sellers"
  ],
  "comment": "10 pages, 6+ figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Integer partitions which are simultaneously $t$--cores for distinct values of $t$ have attracted significant interest in recent years. When $s$ and $t$ are relatively prime, Olsson and Stanton have determined the size of the maximal $(s,t)$-core $\\kappa_{s,t}$. When $k\\geq 2$, a conjecture of Amdeberhan on the maximal $(2k-1,2k,2k+1)$-core $\\kappa_{2k-1,2k,2k+1}$ has also recently been verified by numerous authors. In this work, we analyze the relationship between maximal $(2k-1,2k+1)$-cores and maximal $(2k-1,2k,2k+1)$-cores. In previous work, the first author noted that, for all $k\\geq 1,$ $$ \\vert \\, \\kappa_{2k-1,2k+1}\\, \\vert = 4\\vert \\, \\kappa_{2k-1,2k,2k+1}\\, \\vert $$ and requested a combinatorial interpretation of this unexpected identity. Here, using the theory of abaci, partition dissection, and elementary results relating triangular numbers and squares, we provide such a combinatorial proof.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-20T00:08:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "combinatorial proof",
      "relationship",
      "elementary results relating triangular numbers",
      "first author",
      "distinct values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150606186N"
    }
  }
}