Permutations fixing a kset
Abstract
Let $i(n,k)$ be the proportion of permutations $\pi\in\mathcal{S}_n$ having an invariant set of size $k$. In this note we adapt arguments of the second author to prove that $i(n,k) \asymp k^{\delta} (1+\log k)^{3/2}$ uniformly for $1\leq k\leq n/2$, where $\delta = 1  \frac{1 + \log \log 2}{\log 2}$. As an application we show that the proportion of $\pi\in\mathcal{S}_n$ contained in a transitive subgroup not containing $\mathcal{A}_n$ is at least $n^{\delta+o(1)}$ if $n$ is even.
 Publication:

arXiv eprints
 Pub Date:
 July 2015
 arXiv:
 arXiv:1507.04465
 Bibcode:
 2015arXiv150704465E
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Group Theory
 EPrint:
 17 pages. This is the final version accepted for publication incorporating the referees' suggestions. This version will differ from the published version