Geodesic
On two classes of permutations with number-theoretic conditions on the lengths of the cycles
Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 881-891

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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 $S_n(\Lambda)$. 
@article{MZM_1997_62_6_a8,
     author = {A. I. Pavlov},
     title = {On two classes of permutations with number-theoretic conditions on the lengths of the cycles},
     journal = {Matemati\v{c}eskie zametki},
     pages = {881--891},
     publisher = {mathdoc},
     volume = {62},
     number = {6},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a8/}
}
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A. I. Pavlov. On two classes of permutations with number-theoretic conditions on the lengths of the cycles. Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 881-891. http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a8/