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On two classes of permutations with number-theoretic conditions on the lengths of the cycles. (English. Russian original) Zbl 0917.05007
Math. Notes 62, No. 6, 739-746 (1997); translation from Mat. Zametki 62, No. 6, 881-891 (1997).
Summary: 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 of the finite set $$S_n(\Lambda)$$.

##### MSC:
 05A15 Exact enumeration problems, generating functions 05A05 Permutations, words, matrices
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##### References:
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