Counting Parabolic Double Cosets in Symmetric Groups

TL;DR

This paper introduces a new formula and an efficient algorithm to count parabolic double cosets in symmetric groups, enabling computations for large n and confirming an asymptotic conjecture.

Contribution

It derives a novel formula and polynomial-time algorithm for counting parabolic double cosets in symmetric groups, extending previous computational limits.

Findings

Computed p_n for n up to 5000

Proved an asymptotic formula for p_n

Confirmed conjecture by Billey et al.

Abstract

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