A240675 Number of partitions p of n such that exactly one number is in both p and its conjugate.

1, 0, 0, 3, 4, 6, 8, 8, 9, 22, 22, 34, 50, 60, 74, 105, 120, 144, 186, 234, 280, 358, 440, 524, 665, 782, 954, 1150, 1354, 1630, 1944, 2258, 2666, 3170, 3728, 4365, 5128, 5976, 6978, 8144, 9488, 10952, 12700, 14716, 16932, 19558, 22434, 25764, 29505, 33782

Offset: 1

Author

Clark Kimberling, Apr 12 2014

Keywords

• nonn
• easy

Comments

Second column of the array at A240181. Multiplicities greater than 1 are not counted; e.g. there is exactly one number that is in both {4,1,1} and {3,1,1,1}.

Examples

			a(6) counts these 6 partitions:  51, 42, 411, 3111, 2211, 21111, of which the respective conjugates are 21111, 2211, 3111, 411, 42, 51.
		

Links

• Manfred Scheucher, Table of n, a(n) for n = 1..65
• Manfred Scheucher, Sage Script

Crossrefs

Cf. A240674, 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}] (* A240674 *)
Table[Count[Map[Length, b[n]], 1], {n, 1, z}]   (* A240675 *)

PARI

conjug(v) = {my(m = matrix(#v, vecmax(v))); for (i=1, #v, for (j=1, v[i], m[i, j] = 1;);); vector(vecmax(v), i, sum(j=1, #v, m[j, i]));}
a(n) = {my(v = partitions(n)); my(nb = 0); for (k=1, #v, if (#setintersect(Set(v[k]), Set(conjug(v[k]))) == 1, nb++);); nb;} \\ Michel Marcus, Jun 02 2015

Extensions

More terms from Manfred Scheucher, Jun 01 2015