A007870 Determinant of character table of symmetric group S_n.

1, 1, 2, 6, 96, 2880, 9953280, 100329062400, 10651768002183168000, 150283391703941024789299200000, 9263795272057860957392207640004657152000000000, 16027108137650009941734148595388542471170145479274004480000000000000

Offset: 0

Author

Peter J. Cameron, Götz Pfeiffer [ goetz(AT)dcs.st-and.ac.uk ]

Keywords

• nonn

Examples

			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
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..18
• 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.

Crossrefs

Row-products of A302246 and A302247.

Cf. A000041, A000142, A006128, A006906, A066186, A066633, A086644, A325501, A325504, A325507, A325536.

Cf. A063073.

Programs

GAP

List(List([0..11],n->Flat(Partitions(n))),Product); # Muniru A Asiru, Dec 21 2018

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 *)

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

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)