A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

1, 1, 1, 3, 5, 11, 23, 48, 101, 208, 434

Offset: 0

Author

Gus Wiseman, Mar 29 2025

Keywords

• nonn
• more

Comments

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

Examples

			The multiset {1,2,2,3,3} can be partitioned into a set of sets with distinct sums in 4 ways:
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
so is counted under a(5).
The multisets counted by A382214 but not by A382216 are:
  {1,1,1,1,2,2,3,3,3}
  {1,1,2,2,2,2,3,3,3}
The a(1) = 1 through a(5) = 11 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
              {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
                       {1,2,3,3}  {1,1,2,3,4}
                       {1,2,3,4}  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}
		

Crossrefs

Twice-partitions of this type are counted by A279785, without distinct sums A358914.

Factorizations of this type are counted by A381633, without distinct sums A050326.

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

The complement is counted by A382202.

Without distinct sums we have A382214, complement A292432.

The case of a unique choice is counted by A382459, without distinct sums A382458.

For Heinz numbers: A293243, A381806, A382075, A382200.

For integer partitions: A381990, A381992, A382077, A382078.

Strong version: A382523, A382430, A381996, A292444.

Normal multiset partitions: A034691, A035310, A255906.

Set systems: A050342, A296120, A318361.

Set multipartitions: A089259, A270995, A296119, A318360.

Cf. A000110, A000670, A007716, A050320, A116540, A255903, A275780, A317532, A326518, A326519, A382429, A382460.

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[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,5}]