{
  "id": "1701.04963",
  "version": "v1",
  "published": "2017-01-18T06:23:41.000Z",
  "updated": "2017-01-18T06:23:41.000Z",
  "title": "On the Existence of Tableaux with Given Modular Major Index",
  "authors": [
    "Joshua P. Swanson"
  ],
  "comment": "22 pages, 5 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index $r$ mod $n$, for all $r$. Our result generalizes the $r=1$ case due essentially to (1974) and proves a recent conjecture due to Sundaram (2016) for the $r=0$ case. A byproduct of the proof is an asymptotic equidistribution result for \"almost all\" shapes. The proof uses a representation-theoretic formula involving Ramanujan sums and normalized symmetric group character estimates. Further estimates involving \"opposite\" hook lengths are given which are well-adapted to classifying which partitions $\\lambda \\vdash n$ have $f^\\lambda \\leq n^d$ for fixed $d$. We also give a new proof of a generalization of the hook length formula due to Fomin-Lulov (1995) for symmetric group characters at rectangles. We conclude with some remarks on unimodality of symmetric group characters.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-18T06:23:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10"
    ],
    "keywords": [
      "modular major index",
      "normalized symmetric group character estimates",
      "asymptotic equidistribution result",
      "standard young tableau",
      "hook length formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}