A237113 - OEIS

A237113

Number of partitions of n such that some part is a sum of two other parts.

0, 0, 0, 0, 1, 1, 3, 3, 8, 10, 17, 22, 37, 47, 71, 91, 133, 170, 236, 301, 408, 515, 686, 860, 1119, 1401, 1798, 2232, 2829, 3495, 4378, 5381, 6682, 8165, 10060, 12238, 14958, 18116, 22018, 26533, 32071, 38490, 46265, 55318, 66193, 78843, 93949, 111503, 132326

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

COMMENTS

These are partitions containing the sum of some 2-element submultiset of the parts, a variation of binary sum-full partitions where parts cannot be re-used, ranked by A364462. The complement is counted by A236912. The non-binary version is A237668. For re-usable parts we have A363225. - Gus Wiseman, Aug 10 2023

LINKS

Table of n, a(n) for n=0..48.

FORMULA

a(n) = A000041(n) - A236912(n).

EXAMPLE

Of the 11 partitions of 6, only these 3 include a part that is a sum of two other parts: [3,2,1], [2,2,1,1], [2,1,1,1,1]. Thus, a(6) = 3.

From Gus Wiseman, Aug 09 2023: (Start)

The a(0) = 0 through a(9) = 10 partitions:

. . . . (211) (2111) (321) (3211) (422) (3321)

(2211) (22111) (431) (4221)

(21111) (211111) (3221) (4311)

(4211) (5211)

(22211) (32211)

(32111) (42111)

(221111) (222111)

(2111111) (321111)

(2211111)

(21111111)

(End)

MATHEMATICA

z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]], Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)

u = PartitionsP[Range[z]] - t (* A237113, Peter J. C. Moses, Feb 03 2014 *)

Table[Length[Select[IntegerPartitions[n], Intersection[#, Total/@Subsets[#, {2}]]!={}&]], {n, 0, 30}] (* Gus Wiseman, Aug 09 2023 *)

CROSSREFS

The complement for subsets is A085489, with re-usable parts A007865.

For subsets of {1..n} we have A088809, with re-usable parts A093971.

The complement is counted by A236912, ranks A364461.

The non-binary complement is A237667, ranks A364531.

The non-binary version is A237668, ranks A364532.

With re-usable parts we have A363225, ranks A364348.

The complement with re-usable parts is A364345, ranks A364347.

These partitions have ranks A364462.

The strict case is A364670, with re-usable parts A363226.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A108917 counts knapsack partitions, ranks A299702.

A323092 counts double-free partitions, ranks A320340.

Cf. A002865, A151897, A237984, A325862, A326083, A363260.

Sequence in context: A141577 A245142 A123315 * A293937 A291857 A300672

Adjacent sequences: A237110 A237111 A237112 * A237114 A237115 A237116

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 04 2014

EXTENSIONS

a(0)=0 prepended by Alois P. Heinz, Sep 17 2023

STATUS

approved