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

Offset: 0

Author

Clark Kimberling, Feb 04 2014

Keywords

• nonn

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

Examples

			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)
		

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.

Programs

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 *)

Formula

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

Extensions

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