{
  "id": "math/0409480",
  "version": "v2",
  "published": "2004-09-24T19:06:47.000Z",
  "updated": "2005-03-02T22:31:32.000Z",
  "title": "Dissecting the Stanley Partition Function",
  "authors": [
    "Alexander Berkovich",
    "Frank G. Garvan"
  ],
  "comment": "13 pages, new theorems, examples and Note added, to appear in JCT(A)",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let p[i](n) denote the number of partitions pi of n such that O(pi) - O(pi') = i mod 4. Here O(pi) denotes the number of odd parts of the partition pi and pi' is the conjugate of pi. R. Stanley [13], [14] derived an infinite product representation for the generating function of p[0](n)-p[2](n). Recently, Holly Swisher[15] employed the circle method to show that limit[n->oo] p[0](n)/p(n) = 1/2 (i) and that for sufficiently large n 2 p[0](n) > p(n), if n=0,1 mod 4, 2 p[0](n) < p(n), otherwise. (ii) In this paper we study even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result |p[0](2n) - p[2](2n)| > |p[0](2n+1) - p[2](2n+1)|, n>0. Two proofs of this surprising inequality are given. The first one uses the Gollnitz-Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-03-02T22:31:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P81",
      "11P82",
      "11P83",
      "05A17",
      "05A19"
    ],
    "keywords": [
      "stanley partition function",
      "jacobis triple product identity",
      "gollnitz-gordon partition theorem",
      "naive upper bound estimates",
      "infinite product representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......9480B"
    }
  }
}