A371059 - OEIS

%I #48 Mar 19 2024 12:43:08

%S 1,1,9,14,22,44,74,160,256,462,817,1494,2543,4427,7699,13352,22616,

%T 38610,65052,110004,182961,305007,503299,830648,1356227,2212790,

%U 3583419,5790836

%N Number of conjugacy classes of pairs of commuting elements in the alternating group A_n.

%C 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}.

%C It is equal to the number of conjugacy classes within the centralizers of class representatives of G.

%C This reformulation was employed in the sequence-generating program.

%C It is also equal to the rank of the modular fusion category Z(Rep(G)), the Drinfeld center of Rep(G).

%C These reformulations are explained in the linked MathOverflow posts.

%D A. Davydov, Bogomolov multiplier, double class-preserving automorphisms, and modular invariants for orbifolds. J. Math. Phys. 55 (2014), no. 9, 092305, 13 pp.

%H Sébastien Palcoux, <a href="https://mathoverflow.net/q/466800/34538">Number of conjugacy classes of pairs of commuting elements</a>, MathOverflow.

%H Sébastien Palcoux, <a href="https://mathoverflow.net/q/466864/34538">Is the rank of an integral MTC an upper bound for the prime factors of its FPdim?</a>, MathOverflow.

%o (GAP)

%o List([1..10],n->Sum(List(ConjugacyClasses(AlternatingGroup(n)),c->NrConjugacyClasses(Centralizer(AlternatingGroup(n),Representative(c))))));

%Y Cf. A000702.

%K nonn,more

%O 1,3

%A _Sébastien Palcoux_, Mar 11 2024