{
  "id": "math/0401012",
  "version": "v1",
  "published": "2004-01-03T03:16:35.000Z",
  "updated": "2004-01-03T03:16:35.000Z",
  "title": "On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5",
  "authors": [
    "Alexander Berkovich",
    "Frank G. Garvan"
  ],
  "comment": "14 pages, 1 figure, 2 tables",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "In a recent study of sign-balanced, labelled posets Stanley [13], introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' the conjugate of pi. In [1] Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod 5), p(n) = p[0](n) + p[2](n), where p[i](n) (i = 0, 2) denotes the number of partitions of n with srank = i (mod 4) and p(n) is the number of unrestricted partitions of n. Andrews asked for a partition statistic that would divide the partitions enumerated by p[i](5n + 4) (i = 0, 2) into five equinumerous classes. In this paper we discuss two such statistics. The first one, while new, is intimately related to the Andrews-Garvan [2] crank. The second one is in terms of the 5-core crank, introduced by Garvan, Kim and Stanton [9]. Finally, we discuss some new formulas for partitions that are 5-cores.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-01-03T03:16:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P81",
      "11P83",
      "05A17",
      "05A19"
    ],
    "keywords": [
      "ramanujans partition congruence modulo",
      "andrews-stanley refinement",
      "integral partition statistic srank"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......1012B"
    }
  }
}