{
  "id": "1312.2170",
  "version": "v1",
  "published": "2013-12-08T03:39:09.000Z",
  "updated": "2013-12-08T03:39:09.000Z",
  "title": "A Class of Kazhdan-Lusztig R-Polynomials and q-Fibonacci Numbers",
  "authors": [
    "William Y. C. Chen",
    "Neil J. Y. Fan",
    "Peter L. Guo",
    "Michael X. X. Zhong"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $S_n$ denote the symmetric group on $\\{1,2,\\ldots,n\\}$. For two permutations $u, v\\in S_n$ such that $u\\leq v$ in the Bruhat order, let $R_{u,v}(q)$ and $\\R_{u,v}(q)$ denote the Kazhdan-Lusztig $R$-polynomial and $\\R$-polynomial, respectively. Let $v_n=34\\cdots n\\, 12$, and let $\\sigma$ be a permutation such that $\\sigma\\leq v_n$. We obtain a formula for the $\\R$-polynomials $\\R_{\\sigma,v_n}(q)$ in terms of the $q$-Fibonacci numbers depending on a parameter determined by the reduced expression of $\\sigma$. When $\\sigma$ is the identity $e$, this reduces to a formula obtained by Pagliacci. In another direction, we obtain a formula for the $\\R$-polynomial $\\R_{e,\\,v_{n,i}}(q)$, where $v_{n,i} = 3 4\\cdots i\\,n\\, (i+1)\\cdots (n-1)\\, 12$. In a more general context, we conjecture that for any two permutations $\\sigma,\\tau\\in S_n$ such that $\\sigma\\leq \\tau\\leq v_n$, the $\\R$-polynomial $\\R_{\\sigma,\\tau}(q)$ can be expressed as a product of $q$-Fibonacci numbers multiplied by a power of $q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-08T03:39:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E15",
      "20F55"
    ],
    "keywords": [
      "kazhdan-lusztig r-polynomials",
      "q-fibonacci numbers",
      "permutation",
      "general context",
      "symmetric group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.2170C"
    }
  }
}