Abstract:
Let $\Lambda$ be an arbitrary set of positive integers and $S_n(\Lambda)$ the set of all permutations of degree $n$ for which the lengths of all cycles belong to the set $\Lambda$. In the paper the asymptotics of the ratio $|S_n(\Lambda)|/n!$ as $n\to\infty$ is studied in the following cases: 1) $\Lambda$ is the union of finitely many arithmetic progressions, 2) $\Lambda$ consists of all positive integers that are not divisible by any number from a given finite set of pairwise coprime positive integers. Here $|S_n(\Lambda)|$ stands for the number of elements in the finite set $S_n(\Lambda)$.
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