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Number of partitions of n such that some part is a sum of two other parts.
65

%I #17 Sep 17 2023 18:43:49

%S 0,0,0,0,1,1,3,3,8,10,17,22,37,47,71,91,133,170,236,301,408,515,686,

%T 860,1119,1401,1798,2232,2829,3495,4378,5381,6682,8165,10060,12238,

%U 14958,18116,22018,26533,32071,38490,46265,55318,66193,78843,93949,111503,132326

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

%C 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

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

%e 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.

%e From _Gus Wiseman_, Aug 09 2023: (Start)

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

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

%e (2211) (22111) (431) (4221)

%e (21111) (211111) (3221) (4311)

%e (4211) (5211)

%e (22211) (32211)

%e (32111) (42111)

%e (221111) (222111)

%e (2111111) (321111)

%e (2211111)

%e (21111111)

%e (End)

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

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

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

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

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

%Y The complement is counted by A236912, ranks A364461.

%Y The non-binary complement is A237667, ranks A364531.

%Y The non-binary version is A237668, ranks A364532.

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

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

%Y These partitions have ranks A364462.

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

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

%Y A108917 counts knapsack partitions, ranks A299702.

%Y A323092 counts double-free partitions, ranks A320340.

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

%K nonn

%O 0,7

%A _Clark Kimberling_, Feb 04 2014

%E a(0)=0 prepended by _Alois P. Heinz_, Sep 17 2023