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Asymptotics of the number of permutations with number-theoretic restrictions on cycle length. (English. Russian original) Zbl 0828.05005
Russ. Acad. Sci., Dokl., Math. 49, No. 2, 380-385 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 335, No. 5, 556-559 (1994).
Let \(\Lambda\) be a set of positive integers, \(S_n(\Lambda)\) the set of all permutations of degree \(n\) having only cycles with lengths in the set \(\Lambda\), and \(a_n(\Lambda)= |S_n(\Lambda)|/n!\), where \(|S_n(\Lambda)|\) is the number of elements in the finite set \(S_n(\Lambda)\). In [Discrete Math. Appl. 2, No. 4, 445-459 (1992); translation from Diskretn. Mat. 3, No. 3, 109-129 (1991; Zbl 0735.05009)] we obtained a number of results on the asymptotic behavior of \(a_n(\Lambda)\) as \(n\to \infty\) subject to the density \(\gamma\) of the set \(\Lambda\). In this note, we indicate the asymptotic behavior of \(a_n(\Lambda)\) as \(n\to \infty\) subject to various restrictions of an arithmetic nature on the set \(\Lambda\).

05A15 Exact enumeration problems, generating functions
05A16 Asymptotic enumeration
20B99 Permutation groups
05A05 Permutations, words, matrices