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A371059
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Number of conjugacy classes of pairs of commuting elements in the alternating group A_n.
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0
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1, 1, 9, 14, 22, 44, 74, 160, 256, 462, 817, 1494, 2543, 4427, 7699, 13352, 22616, 38610, 65052, 110004, 182961, 305007, 503299, 830648, 1356227, 2212790, 3583419, 5790836
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OFFSET
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1,3
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COMMENTS
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The number of conjugacy classes of pairs of commuting elements in a finite group G is the cardinality of the set {c(a,b) | a,b in G and ab=ba} where c(a,b) = {(gag^(-1),gbg^(-1)) | g in G}.
It is equal to the number of conjugacy classes within the centralizers of class representatives of G.
This reformulation was employed in the sequence-generating program.
It is also equal to the rank of the modular fusion category Z(Rep(G)), the Drinfeld center of Rep(G).
These reformulations are explained in the linked MathOverflow posts.
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REFERENCES
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A. Davydov, Bogomolov multiplier, double class-preserving automorphisms, and modular invariants for orbifolds. J. Math. Phys. 55 (2014), no. 9, 092305, 13 pp.
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LINKS
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PROG
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(GAP)
List([1..10], n->Sum(List(ConjugacyClasses(AlternatingGroup(n)), c->NrConjugacyClasses(Centralizer(AlternatingGroup(n), Representative(c))))));
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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