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

1,4

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}.

Links

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

Manfred Scheucher, Sage Script

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A240674, A240181.

Sequence in context: A368152 A242822 A295996 * A072152 A199015 A196098

Adjacent sequences: A240672 A240673 A240674 * A240676 A240677 A240678

Keyword

nonn,easy

Author

Clark Kimberling, Apr 12 2014

Extensions

More terms from Manfred Scheucher, Jun 01 2015

Status

approved