A382459 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums in exactly one way.

1, 1, 0, 2, 1, 3, 2, 7, 4, 10, 19

Offset: 0

Author

Gus Wiseman, Apr 01 2025

Keywords

• nonn
• more

Comments

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

Examples

			The normal multiset {1,2,2,2,2,3,3,4} has only one multiset partition into a set of sets with distinct sums: {{2},{1,2},{2,3},{2,3,4}}, so is counted under a(8).
The a(1) = 1 through a(7) = 7 multisets:
  {1}  .  {112}  {1122}  {11123}  {111233}  {1111234}
          {122}          {12223}  {122233}  {1112223}
                         {12333}            {1112333}
                                            {1222234}
                                            {1222333}
                                            {1233334}
                                            {1234444}
		

Crossrefs

Twice-partitions of this type are counted by A279785, A270995, A358914.

Factorizations of this type are counted by A381633, A050320, A050326.

Normal multiset partitions of this type are A381718, A116540, A116539.

Multiset partitions of this type are ranked by A382201, A302478, A302494.

For at least one choice: A382216 (strict A382214), complement A382202 (strict A292432).

For the strong case see: A382430 (strict A292444), complement A382523 (strict A381996).

Without distinct sums we have A382458.

For integer partitions we have A382460, ranks A381870, strict A382079, ranks A293511.

Set multipartitions: A089259, A296119, A318360.

Normal multiset partitions: A034691, A035310, A255906.

Set systems: A050342, A296120, A318361.

Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).

Cf. A000110, A000670, A007716, A255903, A275780, A317532, A321469, A326519, A381078, A381441, A382428.

Programs

Mathematica

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]]==1&]],{n,0,5}]