Permutations fixing a k-set

@article{Eberhard2015PermutationsFA,
  title={Permutations fixing a k-set},
  author={S. Eberhard and K. Ford and B. Green},
  journal={arXiv: Combinatorics},
  year={2015}
}
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. 
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