{
  "id": "1712.10072",
  "version": "v1",
  "published": "2017-12-28T22:31:43.000Z",
  "updated": "2017-12-28T22:31:43.000Z",
  "title": "On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts",
  "authors": [
    "Anthony Zaleski",
    "Doron Zeilberger"
  ],
  "comment": "12 pages, accompanied by Maple package",
  "categories": [
    "math.CO"
  ],
  "abstract": "Tewodros Amdeberhan and Armin Straub initiated the study of enumerating subfamilies of the set of (s,t)-core partitions. While the enumeration of (n+1,n+2)-core partitions into distinct parts is relatively easy (in fact it equals the Fibonacci number F_{n+2}), the enumeration of (n+1,n+2)-core partitions into odd parts remains elusive. Straub computed the first eleven terms of that sequence, and asked for a \"formula,\" or at least a fast way, to compute many terms. While we are unable to find a \"fast\" algorithm, we did manage to find a \"faster\" algorithm, which enabled us to compute 23 terms of this intriguing sequence. We strongly believe that this sequence has an algebraic generating function, since a \"sister sequence\" (see the article), is OEIS sequence A047749 that does have an algebraic generating function. One of us (DZ) is pledging a donation of 100 dollars to the OEIS, in honor of the first person to generate sufficiently many terms to conjecture (and prove non-rigorously) an algebraic equation for the generating function of this sequence, and another 100 dollars for a rigorous proof of that conjecture. Finally, we also develop algorithms that find explicit generating functions for other, more tractable, families of (n+1,n+2)-core partitions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-28T22:31:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "05A15",
      "05A16",
      "05E10"
    ],
    "keywords": [
      "partitions",
      "intriguing problem",
      "algebraic generating function",
      "oeis sequence a047749",
      "odd parts remains"
    ],
    "tags": [
      "research tool"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}