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

0,3

Comments

First column of the array at A240181.

Links

Table of n, a(n) for n=0..54.

Formula

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

Example

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

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 *)

Crossrefs

Cf. A240675, A114701, A240181.

Sequence in context: A010578 A328583 A355537 * A005866 A125584 A230447

Adjacent sequences: A240671 A240672 A240673 * A240675 A240676 A240677

Keyword

nonn,easy

Author

Clark Kimberling, Apr 12 2014

Extensions

Name corrected by Clark Kimberling, Sep 28 2023

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

Status

approved