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Ewens Sampling and Invariable Generation

Published online by Cambridge University Press:  12 April 2018

GERANDY BRITO ,
CHRISTOPHER FOWLER ,
MATTHEW JUNGE  and
AVI LEVY
GERANDY BRITO
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry St, Atlanta, GA 30309, USA (e-mail: gbrito3@gatech.edu)
CHRISTOPHER FOWLER
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry St, Atlanta, GA 30309, USA (e-mail: gbrito3@gatech.edu)
MATTHEW JUNGE
Affiliation:
Department of Mathematics, Duke University, Durham, NC 27708, USA (e-mail: jungem@math.duke.edu)
AVI LEVY
Affiliation:
Microsoft Corporation, One Microsoft Way, Redmond, WA 98052, USA (e-mail: me@avilevy.org)
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Abstract

We study the number of random permutations needed to invariably generate the symmetric group Sn when the distribution of cycle counts has the strong α-logarithmic property. The canonical example is the Ewens sampling formula, for which the special case α = 1 corresponds to uniformly random permutations.

For strong α-logarithmic measures and almost every α, we show that precisely ⌈(1−αlog2)−1⌉ permutations are needed to invariably generate Sn with asymptotically positive probability. A corollary is that for many other probability measures on Sn no fixed number of permutations will invariably generate Sn with positive probability. Along the way we generalize classic theorems of Erdős, Tehran, Pyber, Łuczak and Bovey to permutations obtained from the Ewens sampling formula.

Keywords

Primary 60C05Secondary 12Y0568W2068W3068W40

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 6 , November 2018 , pp. 853 - 891
Copyright
Copyright © Cambridge University Press 2018 

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