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A192983 a(n) is the number of pairs (g, h) of elements of the symmetric group S_n such that g and h have conjugates that commute. 1
1, 4, 24, 264, 5640, 151200, 5722920, 282868992, 18371308032, 1504791561600, 148978034686800, 18007146260231040, 2528615024682544512, 426310052282058252672, 81830910530970671616000, 18305445786667543107072000, 4570435510076312321728158720 (list; graph; refs; listen; history; text; internal format)



a(n) / n!^2 is the probability that two permutation in S_n, chosen independently and uniformly at random, have conjugates that commute.

Apparently n | a(n), and, for  n>1,  n*(n-1) | a(n). - Alexander R. Povolotsky, Sep 30 2011


Table of n, a(n) for n=1..17.

Simon R. Blackburn, John R. Britnell, and Mark Wildon, The probability that a pair of elements of a finite group are conjugate, arXiv:1108.1784, 2011

J. R. Britnell and M. Wildon, Commuting elements in conjugacy classes: an application of Hall's Marriage Theorem to group theory, J. Group Theory, 12 (2009), 795-802.

Mark Wildon, Haskell source code for computing values of the sequence.


For n = 3 the probability that two elements of S_3 have conjugates that commute is a(3)/3!^2 = 2/3. Proof: only the transpositions and three cycles fail to have conjugates that commute; the probability of choosing one permutation from each of these classes is 2*1/2*1/3 = 1/3.


(Haskell) See links for code.


Cf. A087132 (the sum of squares of the sizes of the conjugacy classes of S_n).

Sequence in context: A095340 A141014 A340023 * A077700 A080489 A101228

Adjacent sequences:  A192980 A192981 A192982 * A192984 A192985 A192986




Mark Wildon, Aug 03 2011



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Last modified January 27 16:10 EST 2022. Contains 350608 sequences. (Running on oeis4.)