A382460 - OEIS

A382460

Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 10, 13, 15, 22, 20, 32, 32, 43, 49, 65, 64, 92, 96, 121, 140, 173, 192

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OFFSET

0,5

LINKS

Table of n, a(n) for n=0..27.

EXAMPLE

The partition y = (3,3,2,1,1,1) has 2 partitions into sets: {{1},{3},{1,2},{1,3}} and {{1},{1,3},{1,2,3}}, but only the latter has distinct sums, so y is counted under a(11)

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

1 2 3 4 5 6 7 8 9 A

211 221 411 322 332 441 433

311 2211 331 422 522 442

511 611 711 622

3311 42111 811

32111 3322

4411

32221

43111

52111

MATHEMATICA

hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];

ssfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#, d]&) /@ Select[ssfacs[n/d], Min@@#>d&], {d, Select[Rest[Divisors[n]], SquareFreeQ]}]];

Table[Length[Select[IntegerPartitions[n], Length[Select[ssfacs[Times@@Prime/@#], UnsameQ@@hwt/@#&]]==1&]], {n, 0, 15}]

CROSSREFS

Twice-partitions of this type are counted by A279785.

Multiset partitions of this type are counted by A381633.

Normal multiset partitions of this type are counted by A381718.

These partitions are ranked by A381870.

For no choices we have A381990, ranks A381806, see A382078, ranks A293243.

For at least one choice we have A381992, ranks A382075, see A382077, ranks A382200.

For distinct blocks instead of block-sums we have A382079, ranks A293511.

MM-numbers of these multiset partitions are A382201, see A302478.

For constant instead of strict blocks we have A382301, ranks A381991.

Set systems: A050320, A050326, A050342, A116539, A296120, A318361.

Set multipartitions: A089259, A116540, A270995, A296119, A318360.

A000041 counts integer partitions, strict A000009.

A265947 counts refinement-ordered pairs of integer partitions.

Cf. A002846, A047966, A213427, A299202, A317142, A358914, A381441, A381454, A381636.

Sequence in context: A099072 A382079 A257241 * A239964 A290585 A106464

Adjacent sequences: A382457 A382458 A382459 * A382461 A382462 A382463

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Mar 29 2025

STATUS

approved