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Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation.
2

%I #20 Nov 28 2019 07:16:32

%S 1,1,1,9,22,77,400,2624,20747,183544,1826374,20045348,240262047,

%T 3120641718,43665293393,654731266933,10472819759734,178001257647196,

%U 3203520381407270,60859480965537820,1217072840308660049

%N Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation.

%H Derek Lim, <a href="/A327150/b327150.txt">Table of n, a(n) for n = 0..61</a>

%H MathOverflow, <a href="http://mathoverflow.net/questions/41337/a-general-formula-for-the-number-of-conjugacy-classes-of-mathbbs-n-times-m/">A general formula for the number of conjugacy classes of S_n×S_n acted on by S_n</a>

%F a(n) = (n!/2) * Sum_{K conjugacy class in A_n} 1/|K|.

%e For n = 3, representatives of the n=9 orbits are (e,e), (e,(123)), (e,(132)), ((123),e), ((132),e), ((123),(123)), ((123),(132)), ((132),(123)), ((132),(132)), where e is the identity.

%o (GAP) G:= AlternatingGroup(n);; Size(G)*Sum(List(ConjugacyClasses(G), K -> 1/Size(K)));

%Y Cf. A000702, A110143, A327014, A327151.

%K nonn

%O 0,4

%A _Derek Lim_, Aug 23 2019