Counting Parabolic Double Cosets in Symmetric Groups

Billey, Konvalinka, Petersen, Solfstra, and Tenner recently presented a method for counting parabolic double cosets in Coxeter groups, used it to compute $p_n$, the number of $S_n$, $n\leq13$. In this paper, we derive new formula $p_n$ an efficient polynomial time algorithm evaluating formula. We use these results $n\leq5000$ prove asymptotic that was conjectured by Billey et al.