A240674 Number of partitions p of n that are disjoint from their conjugate.

1, 0, 2, 2, 2, 2, 4, 4, 8, 10, 10, 14, 18, 18, 26, 30, 36, 44, 60, 64, 82, 96, 114, 130, 164, 176, 222, 248, 296, 338, 406, 450, 550, 620, 726, 816, 968, 1074, 1270, 1418, 1648, 1836, 2150, 2382, 2758, 3080, 3534, 3942, 4538, 5034, 5778, 6416, 7312, 8136, 9258

Offset: 0

Author

Clark Kimberling, Apr 12 2014

Keywords

• nonn
• easy

Comments

First column of the array at A240181.

Examples

			a(6) counts these 4 partitions:  6, 33, 222, 111111, of which the respective conjugates are 111111, 222, 33, 6.
		

Crossrefs

Cf. A240675, A114701, A240181.

Programs

Mathematica

z = 30; p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; c[p_] := c[p] = Table[Count[#, ?(# >= i &)], {i, First[#]}] &[p]; b[n] := b[n] = Table[Intersection[p[n, k], c[p[n, k]]], {k, 1, PartitionsP[n]}]; Table[Count[Map[Length, b[n]], 0], {n, 1, z}] (* this sequence *)
Table[Count[Map[Length, b[n]], 1], {n, 1, z}]   (* A240675 *)

Formula

a(n) = 2*A114701(n), for n >= 1.

Extensions

Name corrected by Clark Kimberling, Sep 28 2023

a(0)=1 prepended by Alois P. Heinz, Jul 19 2024