Permutations with equal orders
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Published:2021-01-27
Issue:5
Volume:30
Page:800-810
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ISSN:0963-5483
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Container-title:Combinatorics, Probability and Computing
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language:en
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Short-container-title:Combinator. Probab. Comp.
Author:
Acan Huseyin,Burnette Charles,Eberhard Sean,Schmutz Eric,Thomas James
Abstract
AbstractLet ${\mathbb{P}}(ord\pi = ord\pi ')$ be the probability that two independent, uniformly random permutations of [n] have the same order. Answering a question of Thibault Godin, we prove that ${\mathbb{P}}(ord\pi = ord\pi ') = {n^{ - 2 + o(1)}}$ and that ${\mathbb{P}}(ord\pi = ord\pi ') \ge {1 \over 2}{n^{ - 2}}lg*n$ for infinitely many n. (Here lg*n is the height of the tallest tower of twos that is less than or equal to n.)
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science
Reference18 articles.
1. [13] Pemantle, R. and Wilson, M. C. (2013) Analytic Combinatorics in Several Variables, Vol. 140 of Cambridge Studies in Advanced Mathematics. Cambridge University Press.
2. [15] Thibo (2016) ‘What is the probability that two random permutations have the same order?’, math overflow. http://mathoverflow.net/a/230276
3. [9] Granville, A. The anatomy of integers and permutations. https://www.dms.umontreal.ca/˜andrew/MSI/AnatomyForTheBook.pdf
4. On some problems of a statistical group-theory. III
5. [18] Wilf, H. S. (1986) The asymptotics of e P(z) and the number of elements of each order in S n . Bull. Amer. Math. Soc. (N.S.) 15 228–232.