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

Offset: 0

Author

Gus Wiseman, Mar 29 2025

Keywords

• nonn
• more

Examples

			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
		

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.

Programs

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}]