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%I #16 Apr 25 2022 08:23:16
%S 1,1,6,864,43200000,272097792000000000,
%T 3416681839784939886182400000000000,
%U 1847600699255039694224318542233446367734016245760000000000000000
%N Product of the sizes of the conjugacy classes of the symmetric group S_n.
%H Alois P. Heinz, <a href="/A064430/b064430.txt">Table of n, a(n) for n = 1..13</a>
%F a(n) = (n!)^A000041(n) / A007870(n)^2.
%e a(3) = 6 because the sizes of the conjugacy classes in S_3 are 1,2,3 and the product is 6.
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, [1$2], ((f, g)->
%p [f[1]+g[1], f[2]*g[2]*i^g[1]])(b(n, i-1), b(n-i, min(n-i, i))))
%p end:
%p a:= n-> n!^combinat[numbpart](n)/b(n$2)[2]^2:
%p seq(a(n), n=1..9); # _Alois P. Heinz_, Aug 03 2021
%t b[n_, i_] := b[n, i] = If[n == 0, {1, 1}, Function[{f, g},
%t {f[[1]] + g[[1]], f[[2]]*g[[2]]*i^g[[1]]}][If[i < 2, {0, 1},
%t b[n, i-1]], If[i > n, {0, 1}, b[n-i, i]]]];
%t A007870[n_] := b[n, n][[2]];
%t a[n_] := (n!)^PartitionsP[n]/A007870[n]^2;
%t Table[a[n], {n, 1, 9}] (* _Jean-François Alcover_, Apr 25 2022, after _Alois P. Heinz_ *)
%o (Magma) [ &*[ c[2] : c in ClassesData(Sym(n))] : n in [1..10]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
%Y Cf. A007870, A000041, A063073, A000142.
%K nonn
%O 1,3
%A Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Sep 30 2001
%E More terms from _Vladeta Jovovic_, Oct 04 2001
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