Rough permutations with a fixed set of a given size

Let $r, k, n$ be integers satisfying 1 r k n/2 . Let Rᵣ(n, k) denote the proportion of permutations π ∈ Sₙ 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 Rᵣ(n, k) uniformly for all 2 r k n/2 . This result generalises the corresponding estimate of Eberhard, Ford, and Green for the case $r=1$ .