Permutations fixing a k-set

TL;DR

This paper estimates the proportion of permutations with a fixed-size invariant set and explores implications for the structure of permutation groups, revealing asymptotic behaviors and subgroup containment probabilities.

Contribution

It provides a uniform asymptotic estimate for the proportion of permutations fixing a k-set, extending previous arguments and applying results to subgroup containment probabilities.

Findings

Proportion of permutations fixing a k-set behaves as k^{- ext{delta}} (1+ ext{log} k)^{-3/2}.

At least n^{- ext{delta}+o(1)} of permutations are in transitive subgroups not containing A_n.

Asymptotic estimates hold uniformly for 1 ≤ k ≤ n/2.

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.