{
  "id": "math/0602362",
  "version": "v4",
  "published": "2006-02-16T17:50:36.000Z",
  "updated": "2007-04-28T04:14:48.000Z",
  "title": "The BG-rank of a partition and its applications",
  "authors": [
    "Alexander Berkovich",
    "Frank G. Garvan"
  ],
  "comment": "20 pages. This version has an expanded section 7, where we defined gbg-rank and stated a number of appealing results. We added a new reference. This paper will appear in Adv. Appl. Math",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let \\pi be a partition. In [2] we defined BG-rank(\\pi) as an alternating sum of parities of parts. This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let p_j(n)(a_{t,j}(n)) denote a number of partitions (t-cores) of n with BG-rank=j. Here, we provide an elegant combinatorial proof that 5|p_j(5n+4) by showing that the residue of the 5-core crank mod 5 divides the partitions enumerated by p_j(5n+4) into five equal classes. This proof uses the orbit construction in [2] and new identity for BG-rank. In addition, we find eta-quotient representation for the generating functions for coefficients a_{t,floor((t+1)/4)}(n), a_{t,-floor((t-1)/4)}(n) when t is an odd, positive integer. Finally, we derive explicit formulas for the coefficients a_{5,j}(n) with j=0,1,-1.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2007-04-28T04:14:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P81",
      "11P83",
      "05A17",
      "05A19"
    ],
    "keywords": [
      "applications",
      "elegant combinatorial proof",
      "partition congruence",
      "famous ramanujan modulo",
      "crank mod"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......2362B"
    }
  }
}