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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 $$\Lambda$$.

##### MSC:
 05A15 Exact enumeration problems, generating functions 05A16 Asymptotic enumeration 20B99 Permutation groups 05A05 Permutations, words, matrices
##### Keywords:
number of permutations; cycle length; asymptotic behavior