A381990 Number of integer partitions of n that cannot be partitioned into a set (or multiset) of sets with distinct sums.

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 58, 76, 97, 127, 168, 208, 267, 343, 431, 536, 676, 836, 1045, 1283, 1582, 1949, 2395, 2895, 3549, 4298, 5216, 6281, 7569, 9104, 10953, 13078, 15652, 18627, 22207, 26325, 31278, 37002, 43708, 51597, 60807, 71533, 84031

Offset: 0

Author

Gus Wiseman, Mar 15 2025

Keywords

• nonn

Examples

			The partition y = (3,3,3,2,2,1,1,1,1) has only one multiset partition into a set of sets, namely {{1},{3},{1,2},{1,3},{1,2,3}}, but this does not have distinct sums, so y is counted under a(17).
The a(2) = 1 through a(8) = 9 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (1111)  (11111)  (222)     (4111)     (2222)
                                (3111)    (22111)    (5111)
                                (21111)   (31111)    (22211)
                                (111111)  (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
		

Crossrefs

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

Twice-partitions of this type are counted by A279785.

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

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

These partitions are ranked by A381806, zeros of A381634 and A381633.

The complement is counted by A381992, ranked by A382075.

For distinct blocks we have A382078, complement A382077, unique A382079.

MM-numbers of these multiset partitions (strict blocks with distinct sum) are A382201.

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.

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