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

The strict case is counted by A364272.

The binary complement with re-usable parts version is A364345, A237113, ranks A364347A364462.

The binary version is A237113, ranks A364462).

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

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

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

Cf. A002865, `A007865, ~A085489, A088809, `A093971, A237984, ~A301900, A325862, A326083, A363226, ~A363260, `A364349, `A364350, A364670.

Cf. A237667, A179009.

The binary complement is A236912 (, ranks A364461), with re-usable A364345 (ranks A364347).

The binary version is A237113 (ranks A364462), complement with re-usable A363225 (parts is A364345, ranks A364348)A364347.

The binary version is A237113, ranks A364462).

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

Cf. A002865, `A007865, ~A085489, A088809, `A093971, A237984, ~A301900, A325862, A326083, A363226, ~A363260, `A364349, `A364350, A364670.

Cf. A007865, A085489, A088809, A093971, A301900, A364350.

From Gus Wiseman, Aug 12 2023: (Start) The a(0) = 0 through a(9) = 13 partitions:

The a(0) = 0 through a(9) = 13 partitions:

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

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. - Gus Wiseman, Aug 12 2023

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)

Table[Length[Select[IntegerPartitions[n], Intersection[#, Total/@Subsets[#, {2, Length[#]}]]!={}&]], {n, 0, 15}] (* Gus Wiseman, Aug 12 2023 *)

The binary complement is A236912 (ranks A364461), with re-usable A364345 (ranks A364347).

The binary version is A237113 (ranks A364462), with re-usable A363225 (ranks A364348).

The complement is counted by A237667, ranks A364531.

The strict case is counted by A364272.

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, A237984, A325862, A326083, A363226, A363260, A364349, A364670.

Cf. A007865, A085489, A088809, A093971, A301900, A364350.

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editing

proposed

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