A237668 Number of partitions of n such that some part is a sum of two or more other parts.

0, 0, 0, 0, 1, 1, 4, 4, 10, 13, 23, 27, 49, 60, 93, 115, 170, 210, 300, 370, 510, 632, 846, 1031, 1359, 1670, 2159, 2630, 3355, 4082, 5130, 6220, 7739, 9360, 11555, 13889, 16991, 20402, 24824, 29636, 35855, 42707, 51309, 60955, 72896, 86328, 102826, 121348

Offset

0,7

Comments

These are partitions containing the sum of some non-singleton submultiset of the parts, a variation of non-binary sum-full partitions where parts cannot be re-used, ranked by A364532. The complement is counted by A237667. The binary version is A237113, or A363225 with re-usable parts. This sequence is weakly increasing. - Gus Wiseman, Aug 12 2023

Links

Giovanni Resta, Table of n, a(n) for n = 0..100

Giovanni Resta, C program for computing a(0)-a(100)

Example

a(6) = 4 counts these partitions: 123, 1113, 1122, 11112.
From Gus Wiseman, Aug 12 2023: (Start)
The a(0) = 0 through a(9) = 13 partitions:
  .  .  .  .  (211)  (2111)  (321)    (3211)    (422)      (3321)
                             (2211)   (22111)   (431)      (4221)
                             (3111)   (31111)   (3221)     (4311)
                             (21111)  (211111)  (4211)     (5211)
                                                (22211)    (32211)
                                                (32111)    (33111)
                                                (41111)    (42111)
                                                (221111)   (222111)
                                                (311111)   (321111)
                                                (2111111)  (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
(End)

Mathematica

z = 20; m = Map[Count[Map[MemberQ[#, Apply[Alternatives, Map[Apply[Plus, #] &, DeleteDuplicates[DeleteCases[Subsets[#], _?(Length[#] < 2 &)]]]]] &, IntegerPartitions[#]], False] &, Range[z]]; PartitionsP[Range[z]] - m
(* Peter J. C. Moses, Feb 10 2014 *)
Table[Length[Select[IntegerPartitions[n], Intersection[#, Total/@Subsets[#, {2, Length[#]}]]!={}&]], {n, 0, 15}] (* Gus Wiseman, Aug 12 2023 *)

Crossrefs

Cf. A179009.

The binary complement is A236912, ranks A364461.

The binary version is A237113, ranks A364462.

The complement is counted by A237667, ranks A364531.

The binary version with re-usable parts is A363225, ranks A364348.

The strict case is A364272.

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

These partitions have ranks A364532.

For subsets instead of partitions we have A364534, complement A151897.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A108917 counts knapsack partitions, ranks A299702.

A299701 counts distinct subset-sums of prime indices.

A323092 counts double-free partitions, ranks A320340.

Cf. A002865, A088809, A237984, A325862, A326083, A363226, A364670.

Sequence in context: A182699 A058596 A180964 * A366754 A209423 A357620

Adjacent sequences: A237665 A237666 A237667 * A237669 A237670 A237671

Keyword

nonn

Author

Clark Kimberling, Feb 11 2014

Extensions

a(21)-a(47) from Giovanni Resta, Feb 22 2014

Status

approved