Subgroup generated by specific simple transpositions

Question

Let $I \subseteq \{ 1, 2, \dots, n \}$ and let $W_I$ be the subgroup of the symmetric group $S_n$ generated by the simple transpositions $s_j = (j, j + 1)$ for $j \notin I$. I'm interested in knowing the size of $W_i$ for $1 \le i \le n - 1$, or more specifically the size of the quotient $$W_i/W_{i,i+a}$$ for $1 \le a \le n - i - 1$. I'm also interested in knowing if it's easy to figure out good coset representatives for $W_{i,i+a}$ in $W_i$, e.g. ones of minimal length. Thanks for help or advice on any of these issues.