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The partition y = (5,3,3,1) has no non-singleton submultiset with sum in y, so is counted under a(12).

Cf. A179009.

Cf. A002865, A007865, A237984, `A275972, A179009, A325862, `A326083, A363225, A363226, `A364348, A364350, A364670.

Counts Includes all knapsack partitions (A108917), but first differs at a(12) = 28, A108917(12) = 25. The difference is accounted for by the non-knapsack partitions: (4332), (5331), (33222).

These are partitions not containing the sum of any non-singleton submultiset of the parts, a variation of non-binary sum-free partitions where parts cannot be re-used, ranked by A364531. The complement is counted by A237668. The binary version is A236912. For re-usable parts we have A364350.

The strict case is A364349, binary A364670A364533.

A007865 counts binary sum-free sets w/ re-usable parts, complement A093971.

Cf. A002865, `A025065, ~A026905, ~A007865, A237984, `A275972, A325862, `A326083, A363225, A363226, ~A363260, ~A364347, `A364348, A364350, A364670.

First differs from Counts all knapsack partitions (A108917 ), but first differs at a(12) = 28, A108917(12) = 25. The difference is accounted for by the non-knapsack partitions : (4332), (5331), (33222).

These are partitions not containing the sum of any non-singleton submultiset of the parts, a variation of non-binary sum-free partitions where parts cannot be re-used. The complement is counted by A237668. The binary (2-ary) version is A236912. For re-usable parts we have A364350. [The binary version for re-usable parts is A364345.]

Includes all knapsack partitions (A108917).

Binary version for subsets w/ re-usable parts: A007865, complement A093971.

The strict case is A364349, binary A364670, with re-usable parts A363226.

A007865 counts binary sum-free sets w/ re-usable parts, complement A093971.

Cf. A002865, `A025065, ~A026905, ~A237984, `A275972, A325862, `A326083, A363225, A363226, ~A363260, ~A364347, `A364348, A364350.

Cf. A237668, A179009.

The binary Binary version for subsets is A085489, with w/ re-usable parts : A007865 (, complement A093971).

For subsets of {1..n} we have A151897, complement A364534 (binary A088809)A085489.

The binary version is A236912 (, ranks A364461), complement A237113 (ranks A364462).

The binary complement is A237113, ranks A364462.

`The binary complement version with re-usable parts is counted by A363225, ranks A364348A364345, strict A364346.

`The binary version with re-usable parts is A364345, ranks A364347, strict A364346.

`Counting all linear combination of parts gives A364350.

The complement for subsets is A364534, binary A088809.

Cf. A002865, `A025065, ~A026905, ~A237984, `A275972, A325862, `A326083, A363225, ~A363260, ~A364347, `A364348, A364350.

From Gus Wiseman, Aug 09 2023: (Start)

First differs from A108917 at a(12) = 28, A108917(12) = 25. The difference is accounted for by the non-knapsack partitions (4332), (5331), (33222).

These are partitions not containing the sum of any non-singleton submultiset of the parts, a variation of non-binary sum-free partitions where parts cannot be re-used. The complement is counted by A237668. The binary (2-ary) version is A236912. For re-usable parts we have A364350. [The binary version for re-usable parts is A364345.]

Includes all knapsack partitions (A108917).

(End)

From Gus Wiseman, Aug 09 2023: (Start)

The partition y = (5,3,1,1) has submultiset (3,1,1) with sum in y, so is not counted under a(10).

The partition y = (5,3,3,1) has no submultiset with sum in y, so is counted under a(12).

The a(1) = 1 through a(8) = 12 partitions:

(1) (2) (3) (4) (5) (6) (7) (8)

(11) (21) (22) (32) (33) (43) (44)

(111) (31) (41) (42) (52) (53)

(1111) (221) (51) (61) (62)

(311) (222) (322) (71)

(11111) (411) (331) (332)

(111111) (421) (521)

(511) (611)

(2221) (2222)

(4111) (3311)

(1111111) (5111)

(11111111)

(End)

Table[Length[Select[IntegerPartitions[n], Intersection[#, Total/@Subsets[#, {2, Length[#]}]]=={}&]], {n, 0, 15}] (* Gus Wiseman, Aug 09 2023 *)

The binary version for subsets is A085489, with re-usable parts A007865 (complement A093971).

For subsets of {1..n} we have A151897, complement A364534 (binary A088809).

The binary version is A236912 (ranks A364461), complement A237113 (ranks A364462).

The complement is counted by A237668, ranks A364532.

`The binary complement with re-usable parts is counted by A363225, ranks A364348.

`The binary version with re-usable parts is A364345, ranks A364347, strict A364346.

The strict case is A364349, binary A364670, with re-usable parts A363226.

`Counting all linear combination of parts gives A364350.

These partitions have ranks A364531.

A000041 counts 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, `A025065, ~A026905, ~A237984, `A275972, A325862, `A326083, ~A363260.

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