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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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