A381992 Number of integer partitions of n that can be partitioned into sets with distinct sums.

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 77, 100, 134, 170, 217, 282, 360, 449, 571, 719, 899, 1122, 1391, 1727, 2136, 2616, 3209, 3947, 4800, 5845, 7094, 8602, 10408, 12533, 15062, 18107, 21686, 25956, 30967, 36936, 43897, 52132, 61850, 73157, 86466, 101992, 120195

Offset: 0

Author

Gus Wiseman, Mar 16 2025

Keywords

• nonn

Comments

Also the number of integer partitions of n whose Heinz number belongs to A382075 (can be written as a product of squarefree numbers with distinct sums of prime indices).

Examples

			There are 6 ways to partition (3,2,2,1) into sets:
  {{2},{1,2,3}}
  {{1,2},{2,3}}
  {{1},{2},{2,3}}
  {{2},{2},{1,3}}
  {{2},{3},{1,2}}
  {{1},{2},{2},{3}}
Of these, 3 have distinct block sums:
  {{2},{1,2,3}}
  {{1,2},{2,3}}
  {{1},{2},{2,3}}
so (3,2,2,1) is counted under a(8).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)  (3)    (4)      (5)      (6)        (7)        (8)
            (2,1)  (3,1)    (3,2)    (4,2)      (4,3)      (5,3)
                   (2,1,1)  (4,1)    (5,1)      (5,2)      (6,2)
                            (2,2,1)  (3,2,1)    (6,1)      (7,1)
                            (3,1,1)  (4,1,1)    (3,2,2)    (3,3,2)
                                     (2,2,1,1)  (3,3,1)    (4,2,2)
                                                (4,2,1)    (4,3,1)
                                                (5,1,1)    (5,2,1)
                                                (3,2,1,1)  (6,1,1)
                                                           (3,2,2,1)
                                                           (3,3,1,1)
                                                           (4,2,1,1)
                                                           (3,2,1,1,1)
		

Crossrefs

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.

Twice-partitions of this type are counted by A279785.

Multiset partitions of this type are counted by A381633, zeros of A381634.

For constant instead of strict blocks see A381717, A381636, A381635, A381716, A381991.

Normal multiset partitions of this type are counted by A381718, see A116539.

The complement is counted by A381990, ranked by A381806.

These partitions are ranked by A382075.

For distinct blocks instead of sums we have A382077, complement A382078.

For a unique choice we have A382079.

A000041 counts integer partitions, strict A000009.

A050320 counts multiset partitions of prime indices into sets.

A050326 counts multiset partitions of prime indices into distinct sets.

A265947 counts refinement-ordered pairs of integer partitions.

A382201 lists MM-numbers of sets with distinct sums.

Cf. A002846, A047966, A213427, A279786, A299202, A317142, A381870.

Cf. A293243, A293511, A358914, A381078, A381441, A381454.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#], And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,10}]

Extensions

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025