Abstract Let r, k, n be integers satisfying 1 r k n/2. Let {R}ᵣ (n, k) denote the proportion of permutations {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 {R}ᵣ (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.
Guoyou Qian (Fri,) studied this question.
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