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Discrete Mathematics and Applications

Editor-in-Chief: Zubkov, Andrei

6 Issues per year

CiteScore 2016: 0.16

SCImago Journal Rank (SJR) 2016: 0.231
Source Normalized Impact per Paper (SNIP) 2016: 0.552

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Volume 20, Issue 3


A limit theorem for the logarithm of the order of a random A-permutation

A. L. Yakymiv
Published Online: 2010-07-08 | DOI: https://doi.org/10.1515/dma.2010.015


In this article, a random permutation τn is considered which is uniformly distributed on the set of all permutations of degree n whose cycle lengths lie in a fixed set A (the so-called A-permutations). It is assumed that the set A has an asymptotic density σ > 0, and |k: kn, kA, mkA|/nσ 2 as n → ∞ uniformly in m ∈ [n, Cn] for an arbitrary constant C > 1. The minimum degree of a permutation such that it becomes equal to the identity permutation is called the order of permutation. Let Zn be the order of a random permutation τn . In this article, it is shown that the random variable ln Zn is asymptotically normal with mean l(n) = ∑kA(n) ln(k)/k and variance σ ln3(n)/3, where A(n) = {k: kA, kn}. This result generalises the well-known theorem of P. Erdős and P. Turán where the uniform distribution on the whole symmetric group of permutations Sn is considered, i.e., where A is equal to the set of positive integers N.

About the article

Received: 2008-10-11

Published Online: 2010-07-08

Published in Print: 2010-07-01

Citation Information: Discrete Mathematics and Applications, Volume 20, Issue 3, Pages 247–275, ISSN (Online) 1569-3929, ISSN (Print) 0924-9265, DOI: https://doi.org/10.1515/dma.2010.015.

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