On the number of permutations with a finite set of cycle lengths. (English. Russian original) Zbl 0897.05006

Pavlov, A. I.

Proc. Steklov Inst. Math. 207, 233-242 (1995); translation from Tr. Mat. Inst. Steklova 207, 256-267 (1994).

Let \(\Lambda\) be a set of natural numbers, \(S_n (\Lambda)\) the set of all permutations of degree \(n\) having only cycles with lengths from the set \(\Lambda\), and \(| S_n (\Lambda)|\) the number of elements of the finite set \(S_n (\Lambda)\). We study the asymptotic behavior of \(| S_n (\Lambda)|\) as \(n\to\infty\), depending on the structure of a fixed set \(\Lambda\). The corresponding problem was first posed in [E. A. Bender, SIAM Rev. 16, 485-515 (1974; Zbl 0294.05002)]. The case \(|\Lambda |= \infty\) has been investigated in a number of papers by the author. In the present paper we investigate the case \(|\Lambda |<\infty\).
For the entire collection see [Zbl 0833.00028].

MSC:

05A16	Asymptotic enumeration
05A05	Permutations, words, matrices
20B30	Symmetric groups

Keywords:

permutations; asymptotic behavior