A392967 - OEIS

%I #27 Jan 30 2026 12:42:19

%S 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,

%T 38,40,57,85,48,79,147,101,79,139,232,177,172,243,279,412,294,282,673,

%U 661,442,729,941,885,879,1144,1579,1740,1388,1582,2871,2705,1898,3209,4488,4234

%N Number of partitions of n whose content sum is 0.

%C 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.

%C 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.

%H Alois P. Heinz, <a href="/A392967/b392967.txt">Table of n, a(n) for n = 0..160</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Self-ConjugatePartition.html">Self-Conjugate Partition</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau#Diagrams">Young tableau, Diagrams</a>.

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

%o (PARI)

%o cs(p)=sum(i=1, #p, p[i]*(p[i]-2*i+1)/2)

%o a(n) = { my(c=0); forpart(p=n, if(cs(Vecrev(p))==0, c++)); c } \\ _Andrew Howroyd_, Jan 28 2026

%Y Cf. A000041, A000700.

%K nonn

%O 0,9

%A _Richard Stanley_, Jan 28 2026