{
  "id": "1507.04290",
  "version": "v1",
  "published": "2015-07-15T16:42:33.000Z",
  "updated": "2015-07-15T16:42:33.000Z",
  "title": "Sums over simultaneous $(s,t)$-core partitions",
  "authors": [
    "Victor Y. Wang"
  ],
  "comment": "Version for the arXiv, with 31 pages, 3 figures. Shorter version with 25 pages (currently available at \\url{https://www.overleaf.com/read/xmzxdgbdcpnq}), without the distinguished `arXiv' text, to be submitted for publication. Comments welcome on either",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Fix coprime $s,t\\ge1$. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the finitely many simultaneous $(s,t)$-cores have average size $\\frac{1}{24}(s-1)(t-1)(s+t+1)$, and that the subset of self-conjugate cores has the same average (first shown by Chen--Huang--Wang). We similarly prove a recent conjecture of Fayers that the average weighted by an inverse stabilizer---giving the \"expected size of the $t$-core of a random $s$-core\"---is $\\frac{1}{24}(s-1)(t^2-1)$. We also prove Fayers' conjecture that the analogous self-conjugate average is the same if $t$ is odd, but instead $\\frac{1}{24}(s-1)(t^2+2)$ if $t$ is even. In principle, our explicit methods extend to averages of arbitrary powers. The main new observation is that the stabilizers appearing in Fayers' conjectures have simple formulas in Johnson's $z$-coordinates parameterization of $(s,t)$-cores.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-15T16:42:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A17",
      "05E10",
      "05E18"
    ],
    "keywords": [
      "core partitions",
      "conjecture",
      "simultaneous",
      "explicit methods extend",
      "ehrhart reciprocity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150704290W"
    }
  }
}