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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. 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Zhong, Neil J.Y. Fan, Peter L. Guo, William Y.C. Chen","submitted_at":"2013-12-08T03:39:09Z","abstract_excerpt":"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. 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