zbMATH — the first resource for mathematics

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 \(S_n(\Lambda)\).

05A15 Exact enumeration problems, generating functions
05A05 Permutations, words, matrices
Full Text: DOI
[1] J. Riordan,An Introduction to Combinatorial Analysis, Wiley, New York (1958). · Zbl 0078.00805
[2] E. A. Bender, ”Asymptotic methods in enumeration,” SIAM Rev.,16, 485–515, (1974). · Zbl 0294.05002 · doi:10.1137/1016082
[3] A. I. Pavlov, ”The number of permutations with finite set of the lengths of cycles,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],207, 256–267 (1996).
[4] M. P. Mineev and A. I. Pavlov, ”The number of permutations of a special form,”Mat. Sb. [Math. USSR-Sb.],99, No. 3, 468–476 (1976). · Zbl 0389.05013
[5] Yu. V. Bolotnikov, V. N. Sachkov, and V. E. Tarakanov, ”Some classes of random variables on cyclic permutations,”Mat. Sb. [Math. USSR-Sb.],108, No. 1, 91–104 (1979).
[6] A. I. Pavlov, ”The number and cyclic structure of permutations of certain classes,”Mat. Sb. [Math. USSR-Sb.],124, No. 4, 536–556 (1984). · Zbl 0559.20005
[7] V. F. Kolchin, ”The number of permutations with restrictions on the lengths of cycles,”Diskret. Mat. [Discrete Math. Appl.],1 No. 2, 97–109 (1989).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.