Permutations fixing a k-set

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 $i(n,k)\asymp k^{-\delta }(1+\log k{)}^{-3\mathrm{/}2}$ uniformly for $1\le k\le n\mathrm{/}2$, 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.