A130721 Sum of the cubes of the number of standard Young tableaux over all partitions of n.

1, 1, 2, 10, 64, 596, 8056, 130432, 2534960, 59822884, 1718480368, 56754444440, 2110577206816, 87981286785328, 4129351961475872, 218382856010529472, 12813477368159567200, 822337333595479929044, 57213666993723455063392, 4305630141314873304140008

Offset

0,3

Comments

The sum of the zeroth power of the number f(p) of standard Young tableaux gives the partition function (A000041), the sum of the first power of f(p) gives the involution function (A000085), the sum of the squares of f(p) gives the factorial function (A000142), so this sequence is the natural one after them.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..60

Formula

For p a partition of n, let f(p) be the number of standard Young tableaux with shape p. Then a(n) = sum(f(p)^3) where the sum ranges over all partitions p of n.

Example

a(4) = 1^3 + 3^3 + 2^3 + 3^3 + 1^3 because the five partitions of 4 (namely 4, 3+1, 2+2, 2+1+1, 1+1+1+1) have respectively 1, 3, 2, 3, 1 standard Young tableaux.

Mathematica

h[l_] := With[{n=Length[l]}, Sum[i, {i, l}]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, k_, l_] := g[n, i, l, k] = If[n == 0, h[l]^k, If[i < 1, 0, g[n, i - 1, k, l] + If[i > n, 0, g[n - i, i, k, Append[l, i]]]]];
a[n_] := If[n == 0, 1, g[n, n, 3, {}]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 18 2017, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A000085, A000142.

Column k=3 of A208447.

Sequence in context: A323666 A318814 A371547 * A167449 A064170 A151410

Adjacent sequences: A130718 A130719 A130720 * A130722 A130723 A130724

Keyword

nonn

Author

David A. Madore, Jul 03 2007

Extensions

More terms from Alois P. Heinz, Feb 26 2012

Status

approved