On parabolic Kazhdan-Lusztig R-polynomials for the symmetric group

Parabolic $R$-polynomials were introduced by Deodhar as parabolic analogues of ordinary $R$-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic $R$-polynomials for the symmetric group. Let $S_n$ be the symmetric group on $\{1,2,\dots ,n\}$, and let $S=\{s_i\mathrm{\mid }1\le i\le n-1\}$ be the generating set of $S_n$, where for $1\le i\le n-1$, $s_i$ is the adjacent transposition. For a subset $J\subseteq S$, let $(S_n{)}_J$ be the parabolic subgroup generated by $J$, and let $(S_n{)}^J$ be the set of minimal coset representatives for $S_n\mathrm{/}(S_n{)}_J$. For $u\le v\in (S_n{)}^J$ in the Bruhat order and $x\in \{q,-1\}$, let $R_{u,v}^{J,x}(q)$ denote the parabolic $R$-polynomial indexed by $u$ and $v$. Brenti found a formula for $R_{u,v}^{J,x}(q)$ when $J=S\setminus \{s_i\}$, and obtained an expression for $R_{u,v}^{J,x}(q)$ when $J=S\setminus \{s_{i-1},s_i\}$. We introduce a statistic on pairs of permutations in $(S_n{)}^J$ for $J=S\setminus \{s_{i-2},s_{i-1},s_i\}$. Then we give a formula for $R_{u,v}^{J,x}(q)$, where $J=S\setminus \{s_{i-2},s_{i-1},s_i\}$ and $i$ appears after $i-1$ in $v$. We also pose a conjecture for $R_{u,v}^{J,x}(q)$, where $J=S\setminus \{s_k,s_{k+1},\dots ,s_i\}$ with $1\le k\le i\le n-1$ and the elements $k+1,k+2,\dots ,i$ appear in increasing order in $v$.