On Counting Double Centralizers of Symmetric Groups

Abstract

Let $S_{2m}$ be symmetric group, $h_0=(1\ 2)\cdots(2m-1\ 2m)$ and $H=C(h_0)$. We clarify the structure of $gHg^{-1}\cap H, g\in S_{2m}$, and using tools from analytic combinatorics we prove that the permutations $g$ such that $|gHg^{-1}\cap H|$ bounded by $m^{O(1)}$ have density zero.