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A007870 Determinant of character table of symmetric group S_n. 14
1, 1, 2, 6, 96, 2880, 9953280, 100329062400, 10651768002183168000, 150283391703941024789299200000, 9263795272057860957392207640004657152000000000, 16027108137650009941734148595388542471170145479274004480000000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Amritanshu Prasad, Symmetric Functions, Chapter 5, Representation Theory: a Combinatorial Viewpoint, Cambridge Studies in Adv. Math. 147 (2014), p. 107.
F. W. Schmidt and R. Simion, On a partition identity, J. Combin. Theory, A 36 (1984), 249-252.
D. Vaintrob, A product identity for partitions, MathOverflow, June 2012.
FORMULA
Product of all parts of all partitions of n.
From Gus Wiseman, May 09 2019: (Start)
a(n) = A003963(A325501(n)).
A001222(a(n)) = A325536(n).
A001221(a(n)) = A000720(n).
(End)
EXAMPLE
1 + x + 2*x^2 + 6*x^3 + 96*x^4 + 2880*x^5 + 9953280*x^6 + 100329062400*x^7 + ...
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
MAPLE
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1$2], ((f, g)->
[f[1]+g[1], f[2]*g[2]*i^g[1]])(b(n, i-1), b(n-i, min(n-i, i))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..12); # Alois P. Heinz, Jul 30 2013
MATHEMATICA
Needs["Combinatorica`"]; Table[Times@@Flatten[Partitions[n]], {n, 10}]
a[ n_] := If[n < 0, 0, Times @@ Flatten @ IntegerPartitions @ n] (* Michael Somos, Jun 11 2012 *)
Table[Exp[Total[Map[Log, IntegerPartitions [n]], 2]], {n, 1, 25}] (* Richard R. Forberg, Dec 08 2014 *)
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 *)
PROG
(GAP) List(List([0..11], n->Flat(Partitions(n))), Product); # Muniru A Asiru, Dec 21 2018
(Python)
from sympy import prod
from sympy.utilities.iterables import ordered_partitions
a = lambda n: prod(map(prod, ordered_partitions(n))) if n > 0 else 1
print([a(n) for n in range(0, 12)]) # Darío Clavijo, Feb 22 2024
CROSSREFS
Row-products of A302246 and A302247.
Sequence in context: A092287 A035482 A322716 * A081992 A066091 A100704
KEYWORD
nonn
AUTHOR
Peter J. Cameron, Götz Pfeiffer [ goetz(AT)dcs.st-and.ac.uk ]
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)