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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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1569-3929
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Volume 20, Issue 3

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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

Abstract

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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Citing Articles

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[1]
A. L. Yakymiv
Theory of Probability & Its Applications, 2018, Volume 63, Number 2, Page 209
[2]
Арсен Любомирович Якымив and Arsen Lubomirovich Yakymiv
Теория вероятностей и ее применения, 2018, Volume 63, Number 2, Page 260
[3]
Арсен Любомирович Якымив and Arsen Lubomirovich Yakymiv
Дискретная математика, 2017, Volume 29, Number 1, Page 136
[4]
Арсен Любомирович Якымив and Arsen Lubomirovich Yakymiv
Математический сборник, 2016, Volume 207, Number 2, Page 143
[5]
A. L. Yakymiv
Theory of Probability & Its Applications, 2015, Volume 59, Number 1, Page 114
[6]
G. I. Ivchenko and M. V. Soboleva
Discrete Mathematics and Applications, 2011, Volume 21, Number 4

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