Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-30T21:18:41.599Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
CHRISTOPHER FOWLER
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry St, Atlanta, GA 30309, USA (e-mail: [email protected])
MATTHEW JUNGE
Affiliation:
Department of Mathematics, Duke University, Durham, NC 27708, USA (e-mail: [email protected])
AVI LEVY
Affiliation:
Microsoft Corporation, One Microsoft Way, Redmond, WA 98052, USA (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Apostol, T. M. (1976) Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer.Google Scholar
[2] Arratia, R., Barbour, A. and Tavaré, S. (1992) Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2 519–535.Google Scholar
[3] Arratia, R., Barbour, A. and Tavaré, S. (2000) Limits of logarithmic combinatorial structures. Ann. Probab. 28 1620–1644.Google Scholar
[4] Arratia, R., Barbour, A. and Tavaré, S. (2016) Exploiting the Feller coupling for the Ewens sampling formula. Statist. Sci. 31 27–29.Google Scholar
[5] Arratia, R. and Tavaré, S. (1992) The cycle structure of random permutations. Ann. Probab. 20 1567–1591.Google Scholar
[6] Arratia, R. and Tavaré, S. (1994) Independent process approximations for random combinatorial structures. Adv. Math. 104 90–154.Google Scholar
[7] Babai, L. (1981) On the order of uniprimitive permutation groups. Ann. of Math. 113 553–568.Google Scholar
[8] Betz, V., Ueltschi, D. and Velenik, Y. (2011) Random permutations with cycle weights. Ann. Appl. Probab. 21 312–331.Google Scholar
[9] Bovey, J. (1980) The probability that some power of a permutation has small degree. Bull. London Math. Soc. 12 47–51.Google Scholar
[10] Crane, H. (2016) The ubiquitous Ewens sampling formula. Statist. Sci. 31 1–19.Google Scholar
[11] Crane, H. (2016) Rejoinder: The ubiquitous Ewens sampling formula. Statist. Sci. 31 37–39.Google Scholar
[12] Davenport, J. and Smith, G. (2000) Fast recognition of alternating and symmetric Galois groups. J. Pure Appl. Algebra 153 17–25.Google Scholar
[13] Dixon, J. D. (1992) Random sets which invariably generate the symmetric group. Discrete Math. 105 25–39.Google Scholar
[14] Eberhard, S., Ford, K. and Green, B. (2016) Permutations fixing a k-set. Internat. Math. Res. Notices 2016 6713–6731.Google Scholar
[15] Eberhard, S., Ford, K. and Green, B. (2017) Invariable generation of the symmetric group. Duke Math. J. 166 1573–1590.Google Scholar
[16] Ercolani, N. M. and Ueltschi, D. (2014) Cycle structure of random permutations with cycle weights. Random Struct. Alg. 44 109–133.Google Scholar
[17] Erdős, P. and Turán, P. (1967) On some problems of a statistical group-theory, II. Acta Math. Hungar. 18 151–163.Google Scholar
[18] Euler, L. (1737) Observationes circa series infinitas. Commentarii Academiae Scientarum Petropolitanae 9 160–188.Google Scholar
[19] Ewens, W. (1972) The sampling theory of selectively neutral alleles. Theoret. Popul. Biol. 3 87–112.Google Scholar
[20] Feller, W. (1945) The fundamental limit theorems in probability. Bull. Amer. Math. Soc. 51 800–832.Google Scholar
[21] Gladkich, A. and Peled, R. (2018) On the cycle structure of Mallows permutations. Ann. Probab. 46 1114–1169. doi: 10.1214/17-AOP1202.Google Scholar
[22] Granville, A. (2008) The anatomy of integers and permutations. Preprint.Google Scholar
[23] Guralnick, R. and Magaard, K. (1998) On the minimal degree of a primitive permutation group. J. Algebra 207 127–145.Google Scholar
[24] Heintz, J. (1986) On polynomials with symmetric Galois group which are easy to compute. Theoret. Comput. Sci. 47 99–105.Google Scholar
[25] Kenyon, R., Kral, D., Radin, C. and Winkler, P. (2015) Permutations with fixed pattern densities. arXiv:1506.02340Google Scholar
[26] Łuczak, T. and Pyber, L. (1993) On random generation of the symmetric group. Combin. Probab. Comput. 2 505–512.Google Scholar
[27] Maier, H. and Tenenbaum, G. (1984) On the set of divisors of an integer. Inventio. Math. 76 121–128.Google Scholar
[28] Mallows, C. L. (1957) Non-null ranking models, I. Biometrika 44 114–130.Google Scholar
[29] Mukherjee, S. (2016) Fixed points and cycle structure of random permutations. Electron. J. Probab. 21 #40.Google Scholar
[30] Musser, D. R. (1978) On the efficiency of a polynomial irreducibility test. J. Assoc. Comput. Mach. 25 271–282.Google Scholar
[31] Pemantle, R., Peres, Y. and Rivin, I. (2016) Four random permutations conjugated by an adversary generate Sn with high probability. Random Struct. Alg. 49 409–428.Google Scholar
[32] van der Waerden, B. (1934) Die Seltenheit der Gleichungen mit Affekt. Math. Ann. 109 13–16.Google Scholar