{
  "id": "1909.11808",
  "version": "v1",
  "published": "2019-09-25T23:06:31.000Z",
  "updated": "2019-09-25T23:06:31.000Z",
  "title": "Simultaneous core partitions with nontrivial common divisor",
  "authors": [
    "Jean-Baptiste Gramain",
    "Rishi Nath",
    "James A. Sellers"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "A tremendous amount of research has been done in the last two decades on $(s,t)$-core partitions when $s$ and $t$ are positive integers with no common divisor. Here we change perspective slightly and explore properties of $(s,t)$-core and $(\\bar{s},\\bar{t})$-core partitions for $s$ and $t$ with nontrivial common divisor $g$. We begin by revisiting work by D. Aukerman, D. Kane and L. Sze on $(s,t)$-core partitions for nontrivial $g$ before obtaining a generating function for the number of $(\\bar{s},\\bar{t})$-core partitions of $n$ under the same conditions. Our approach, using the $g$-core, $g$-quotient and bar-analogues, allows for new results on $t$-cores and self-conjugate $t$-cores that are {\\it not} $g$-cores and $\\bar{t}$-cores that are {\\it not} $\\bar{g}$-cores, thus strengthening positivity results of K. Ono and A. Granville, J. Baldwin et. al., and I. Kiming. We then detail a new bijection between self-conjugate $(s,t)$-core and $(\\bar{s},\\bar{t})$-core partitions for $s$ and $t$ odd with odd, nontrivial common divisor $g$. Here the core-quotient construction fits remarkably well with certain lattice-path labelings due to B. Ford, H. Mai, and L. Sze and C. Bessenrodt and J. Olsson. Along the way we give a new proof of a correspondence of J. Yang between self-conjugate $t$-core and $\\bar{t}$-core partitions when $t$ is odd and positive. We end by noting $(s,t)$-core and $(\\bar{s}, \\bar{t})$-core partitions inherit Ramanujan-type congruences from those of $g$-core and $\\bar{g}$-core partitions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-25T23:06:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83"
    ],
    "keywords": [
      "nontrivial common divisor",
      "simultaneous core partitions",
      "core partitions inherit ramanujan-type congruences",
      "core-quotient construction fits",
      "self-conjugate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}