{
  "id": "1312.2080",
  "version": "v1",
  "published": "2013-12-07T10:10:49.000Z",
  "updated": "2013-12-07T10:10:49.000Z",
  "title": "k-Marked Dyson Symbols and Congruences for Moments of Cranks",
  "authors": [
    "William Y. C. Chen",
    "Kathy Q. Ji",
    "Erin Y. Y. Shen"
  ],
  "comment": "19 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of $2k$-th symmetrized moment $\\eta_{2k}(n)$ of ranks of partitions of $n$. Recently, Garvan introduced the $2k$-th symmetrized moment $\\mu_{2k}(n)$ of cranks of partitions of $n$ in the study of the higher-order spt-function $spt_k(n)$. In this paper, we give a combinatorial interpretation of $\\mu_{2k}(n)$. We introduce $k$-marked Dyson symbols based on a representation of ordinary partitions given by Dyson, and we show that $\\mu_{2k}(n)$ equals the number of $(k+1)$-marked Dyson symbols of $n$. We then introduce the full crank of a $k$-marked Dyson symbol and show that there exist an infinite family of congruences for the full crank function of $k$-marked Dyson symbols which implies that for fixed prime $p\\geq 5$ and positive integers $r$ and $k\\leq (p-1)/2$, there exist infinitely many non-nested arithmetic progressions $An+B$ such that $\\mu_{2k}(An+B)\\equiv 0\\pmod{p^r}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-07T10:10:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83",
      "05A30"
    ],
    "keywords": [
      "k-marked dyson symbols",
      "congruences",
      "th symmetrized moment",
      "combinatorial interpretation",
      "full crank function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.2080C"
    }
  }
}