On counting centralizer subgroups of symmetric groups

Abstract

Let $S_{2m}$ be the symmetric group, $h=(12)(34)\cdots (2m-12m)$ and $H=C(h)$. We consider the structure of $gH{g}^{-1}\cap H$ for any $g\in S_{2m}$. We prove the permutations $g$ which makes $gH{g}^{-1}\cap H$ have size of polynomial in $m$ have density zero.