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Permutations with equal orders

Part of: Combinatorial probability

Published online by Cambridge University Press:  27 January 2021

Huseyin Acan ,
Charles Burnette ,
Sean Eberhard ,
Eric Schmutz  and
James Thomas
Huseyin Acan
Affiliation:
Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA
Charles Burnette
Affiliation:
Mathematics Department, Xavier University of Louisiana, New Orleans, LA 70125, USA
Sean Eberhard
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WB, UK
Eric Schmutz*
Affiliation:
Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA
James Thomas
Affiliation:
Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA
*
*Corresponding author. Email: schmutze@drexel.edu
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Abstract

Let ${\mathbb{P}}(ord\pi = ord\pi ')$ 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.)

MSC classification

Primary: 60C05: Combinatorial probability

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Type
Paper
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Combinatorics, Probability and Computing , Volume 30 , Issue 5 , September 2021 , pp. 800 - 810
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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