{
  "id": "1706.08631",
  "version": "v1",
  "published": "2017-06-27T00:37:07.000Z",
  "updated": "2017-06-27T00:37:07.000Z",
  "title": "Refined Cyclic Sieving on Words for the Major Index Statistic",
  "authors": [
    "Connor Ahlbach",
    "Joshua Swanson"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action and a polynomial. A key example arises from the length generating function for minimal length coset representatives of a parabolic quotient of a finite Coxeter group. In type A, this result can be phrased in terms of the natural cyclic action on words of fixed content. There is a natural notion of refinement for many CSP's. We formulate and prove a refinement, with respect to the major index statistic, of this CSP on words of fixed content by also fixing the cyclic descent type. The argument presented is completely different from Reiner-Stanton-White's representation-theoretic approach. It is combinatorial and largely, though not entirely, bijective in a sense we make precise with a \"universal\" sieving statistic on words, \"flex\". A building block of our argument involves cyclic sieving for shifted subset sums, which also appeared in Reiner-Stanton-White. We give an alternate, largely bijective proof of a refinement of this result by extending some ideas of Wagon-Wilf.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-27T00:37:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "major index statistic",
      "refined cyclic sieving",
      "finite cyclic group action",
      "minimal length coset representatives",
      "reiner-stanton-whites representation-theoretic approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}