A392967 Number of partitions of n whose content sum is 0.

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 6, 9, 5, 5, 8, 9, 18, 10, 9, 25, 28, 14, 24, 42, 29, 38, 40, 57, 85, 48, 79, 147, 101, 79, 139, 232, 177, 172, 243, 279, 412, 294, 282, 673, 661, 442, 729, 941, 885, 879, 1144, 1579, 1740, 1388, 1582, 2871, 2705, 1898, 3209, 4488, 4234

Offset

0,9

Comments

The content of square (i,j) of the Young diagram of a partition p is j-i. The content sum of p is the sum of its contents.

All self-conjugate partitions have content sum 0. The least n for which there is a non-self-conjugate partition with content sum 0 is 15. This explains why the present sequence agrees with A000700 for n<15.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..160

Eric Weisstein's World of Mathematics, Self-Conjugate Partition.

Wikipedia, Young tableau, Diagrams.

Example

For n=15, there are four self-conjugate partitions together with (6,3,2,2,2) and (5,5,2,1,1,1).

Prog

(PARI)
cs(p)=sum(i=1, #p, p[i]*(p[i]-2*i+1)/2)
a(n) = { my(c=0); forpart(p=n, if(cs(Vecrev(p))==0, c++)); c } \\ Andrew Howroyd, Jan 28 2026

Crossrefs

Cf. A000041, A000700.

Sequence in context: A032229 A024366 A218123 * A288157 A333701 A140682

Adjacent sequences: A392964 A392965 A392966 * A392968 A392969 A392970

Keyword

nonn

Author

Richard Stanley, Jan 28 2026

Status

approved