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%I #71 Mar 03 2024 11:29:38
%S 1,1,2,6,96,2880,9953280,100329062400,10651768002183168000,
%T 150283391703941024789299200000,
%U 9263795272057860957392207640004657152000000000,16027108137650009941734148595388542471170145479274004480000000000000
%N Determinant of character table of symmetric group S_n.
%H Alois P. Heinz, <a href="/A007870/b007870.txt">Table of n, a(n) for n = 0..18</a>
%H Amritanshu Prasad, <a href="https://doi.org/10.1017/CBO9781139976824">Symmetric Functions</a>, Chapter 5, Representation Theory: a Combinatorial Viewpoint, Cambridge Studies in Adv. Math. 147 (2014), p. 107.
%H F. W. Schmidt and R. Simion, <a href="http://dx.doi.org/10.1016/0097-3165(84)90012-8">On a partition identity</a>, J. Combin. Theory, A 36 (1984), 249-252.
%H D. Vaintrob, <a href="http://mathoverflow.net/questions/99271/">A product identity for partitions</a>, MathOverflow, June 2012.
%F Product of all parts of all partitions of n.
%F From _Gus Wiseman_, May 09 2019: (Start)
%F a(n) = A003963(A325501(n)).
%F A001222(a(n)) = A325536(n).
%F A001221(a(n)) = A000720(n).
%F (End)
%e 1 + x + 2*x^2 + 6*x^3 + 96*x^4 + 2880*x^5 + 9953280*x^6 + 100329062400*x^7 + ...
%e The integer partitions of 4 are {(4), (3,1), (2,2), (2,1,1), (1,1,1,1)} with product 4*3*1*2*2*2*1*1*1*1*1*1 = 96. - _Gus Wiseman_, May 09 2019
%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-> b(n$2)[2]:
%p seq(a(n), n=0..12); # _Alois P. Heinz_, Jul 30 2013
%t Needs["Combinatorica`"]; Table[Times@@Flatten[Partitions[n]], {n, 10}]
%t a[ n_] := If[n < 0, 0, Times @@ Flatten @ IntegerPartitions @ n] (* _Michael Somos_, Jun 11 2012 *)
%t Table[Exp[Total[Map[Log, IntegerPartitions [n]], 2]], {n, 1, 25}] (* _Richard R. Forberg_, Dec 08 2014 *)
%t b[n_, i_] := b[n, i] = If[n == 0, {1, 1}, Function[{f, g}, {f[[1]] + g[[1]], f[[2]]*g[[2]]*i^g[[1]]}][If[i < 2, {0, 1}, b[n, i - 1]], If[i > n, {0, 1}, b[n - i, i]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%o (GAP) List(List([0..11],n->Flat(Partitions(n))),Product); # _Muniru A Asiru_, Dec 21 2018
%o (Python)
%o from sympy import prod
%o from sympy.utilities.iterables import ordered_partitions
%o a = lambda n: prod(map(prod, ordered_partitions(n))) if n > 0 else 1
%o print([a(n) for n in range(0, 12)]) # _Darío Clavijo_, Feb 22 2024
%Y Row-products of A302246 and A302247.
%Y Cf. A000041, A000142, A006128, A006906, A066186, A066633, A086644, A325501, A325504, A325507, A325536.
%K nonn
%O 0,3
%A _Peter J. Cameron_, Götz Pfeiffer [ goetz(AT)dcs.st-and.ac.uk ]
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