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A000041 counts integer partitions, strict A000009.

A046663 counts partitions w/o a submultiset summing to k, strict A365663.

A365543 counts partitions with a submultiset summing to k, strict A365661.

Cf. A000041, A018818, A046663, A047967, A276024, A304792, A325799, A365543, A365658, A365918, A365921.

For parts instead of sums we have A000009, (sc. coverings), ranks A055932.

A276024 counts positive subset-sums of partitions, strict A284640.

`A325799 counts non-subset-sums of prime indices.

A364350 counts combination-free strict partitions.

Cf. A018818, A047967, `A264401, `A299701, A276024, A304792, `A364272, `A365545, A325799, A365658, A365918, A365921.

Cf. A002865, A006827, A018818, A047967, A080259, `A264401, `A299701, A304792, `A364272, `A365545, A365658, A365918, A365919, A365921.

a(n) = A000041(n) - A365924(n). - Gus Wiseman, Oct 14 2023

From Gus Wiseman, Oct 14 2023: (Start)

The a(1) = 1 through a(8) = 10 partitions:

(1) (11) (21) (211) (221) (321) (421) (3221)

(111) (1111) (311) (2211) (2221) (3311)

(2111) (3111) (3211) (4211)

(11111) (21111) (4111) (22211)

(111111) (22111) (32111)

(31111) (41111)

(211111) (221111)

(1111111) (311111)

(2111111)

(11111111)

(End)

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]]; Table[Length[Select[IntegerPartitions[n], nmz[#]=={}&]], {n, 0, 15}] (* Gus Wiseman, Oct 14 2023 *)

Cf. A002033, A003513, A188431, A209405, A261036, A286929, A286097.

For parts instead of sums we have A000009, ranks A055932.

The strict case is A188431, complement A365831.

These partitions have ranks A325781.

First column k = 0 of A365923.

The complement is counted by A365924, ranks A365830.

A000041 counts integer partitions, strict A000009.

A046663 counts partitions w/o a submultiset summing to k, strict A365663.

A276024 counts positive subset-sums of partitions, strict A284640.

`A325799 counts non-subset-sums of prime indices.

A364350 counts combination-free strict partitions.

A365543 counts partitions with a submultiset summing to k, strict A365661.

Cf. A002865, A006827, A018818, A047967, A080259, A264401, A299701, A304792, A364272, A365545, A365658, A365918, A365919, A365921.

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Nathaniel Johnston and Sarah Plosker, <a href="https://arxiv.org/abs/2308.15611">Laplacian {-1,0,1}- and {-1,1}-diagonalizable graphs</a>, arXiv:2308.15611 [math.CO], 2023.

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