Rough permutations with a fixed set of a given size

Abstract Let $r, k, n$ be integers satisfying $1\leqslant r\leqslant k\leqslant n/2$ . Let ${{\mathcal{R}}}_r(n, k)$ denote the proportion of permutations $\pi \in {{\mathcal{S}}}_n$ that fix a set of size $k$ and have no cycle of length less than $r$ . In this note, we determine the order of magnitude of ${{\mathcal{R}}}_r(n, k)$ uniformly for all $2\leqslant r\leqslant k\leqslant n/2$ . This result generalises the corresponding estimate of Eberhard, Ford, and Green for the case $r=1$ .