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A192983
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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.
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1
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1, 4, 24, 264, 5640, 151200, 5722920, 282868992, 18371308032, 1504791561600, 148978034686800, 18007146260231040, 2528615024682544512, 426310052282058252672, 81830910530970671616000, 18305445786667543107072000, 4570435510076312321728158720
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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PROG
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(Haskell) -- See links for code.
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CROSSREFS
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Cf. A087132 (the sum of squares of the sizes of the conjugacy classes of S_n).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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