{
  "id": "1408.6484",
  "version": "v1",
  "published": "2014-08-27T17:58:41.000Z",
  "updated": "2014-08-27T17:58:41.000Z",
  "title": "Cyclic Sieving and Plethysm Coefficients",
  "authors": [
    "David B Rush"
  ],
  "comment": "36 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A combinatorial expression for the coefficient of the Schur function $s_{\\lambda}$ in the expansion of the plethysm $p_{n/d}^d \\circ s_{\\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\\lambda$ is rectangular. In these cases, the coefficient $\\langle p_{n/d}^d \\circ s_{\\mu}, s_{\\lambda} \\rangle$ is shown to count, up to sign, the number of fixed points of an $\\langle s_{\\mu}^n, s_{\\lambda} \\rangle$-element set under the $d^{\\text{th}}$ power of an order $n$ cyclic action. If $n=2$, the action is the Sch\\\"utzenberger involution on semistandard Young tableaux (also known as evacuation), and, if $\\lambda$ is rectangular, the action is a certain power of Sch\\\"utzenberger and Shimozono's jeu-de-taquin promotion. This work extends results of Stembridge and Rhoades linking fixed points of the Sch\\\"utzenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carr\\'e and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-27T17:58:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05E10",
      "05E18"
    ],
    "keywords": [
      "plethysm coefficients",
      "cyclic sieving",
      "schur function",
      "domino tableaux rule",
      "ribbon tableaux enumeration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.6484R"
    }
  }
}