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%I #29 Mar 13 2015 15:13:43
%S 1,1,4,30,312,3720,64080,1305360,33949440,1019692800,36360576000,
%T 1487539468800,69633899596800,3649476307276800,213929162589542400,
%U 13848506938506240000,986705192227442688000,76724136092268048384000,6491471142159880740864000
%N Ordered pairs of permutation functions on n elements where f(g(g(x))) = g(g(f(x))).
%C From _Paul Boddington_, Feb 24 2015: (Start)
%C Suppose G is the symmetric group on n letters. For each g in G, the set of f satisfying fgg = ggf is just the centralizer Z_gg(G). However |Z_gg(G)| is clearly constant on conjugacy classes of G. By the orbit-stabilizer theorem the size of the conjugacy class containing g is |G| / |Z_g(G)|. Since |G| = n! and Z_g(G) is a subgroup of Z_gg(G) we see that a(n) equals n! multiplied by the sum of indices |Z_gg(G) : Z_g(G)| where the sum is over representatives of the conjugacy classes of G. Since the conjugacy classes of G correspond to partitions of n (A000041), this makes it relatively easy to find terms.
%C a(n) appears to equal n! * A082733(n).
%C (End)
%D John F. Humphreys, A Course In Group Theory, Oxford Science Publications, 1996, chapter 10.
%H Hiroaki Yamanouchi, <a href="/A239841/b239841.txt">Table of n, a(n) for n = 0..60</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SymmetricGroup.html">Symmetric Group</a>
%Y Cf. A000012, A000085, A000142, A088311, A053529, A001044.
%K nonn
%O 0,3
%A _Chad Brewbaker_, Mar 27 2014
%E a(8)-a(9) from _Giovanni Resta_, Mar 27 2014
%E More terms from _Paul Boddington_, Feb 23 2015
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