A382458 Number of normal multisets of size n that can be partitioned into a set of sets in exactly one way.

1, 1, 0, 2, 1, 3, 0, 7, 3, 11, 18, 9

Offset: 0

Author

Gus Wiseman, Mar 30 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 three multiset partitions into a set of sets:
  {{2},{1,2},{2,3},{2,3,4}}
  {{2},{2,3},{2,4},{1,2,3}}
  {{2},{3},{1,2},{2,3},{2,4}}
so is not counted under a(8).
The a(1) = 1 through a(7) = 7 normal multisets:
  {1}  .  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  .  {1,1,1,1,2,3,4}
          {1,2,2}             {1,2,2,2,3}     {1,1,1,2,2,2,3}
                              {1,2,3,3,3}     {1,1,1,2,3,3,3}
                                              {1,2,2,2,2,3,4}
                                              {1,2,2,2,3,3,3}
                                              {1,2,3,3,3,3,4}
                                              {1,2,3,4,4,4,4}
		

Crossrefs

For constant instead of strict blocks we have A000045.

Factorizations of this type are counted by A050326, with distinct sums A381633.

For the strong case see A292444, A382430, complement A381996, A382523.

MM-numbers of sets of sets are A302494, see A302478, A382201.

Twice-partitions into distinct sets are counted by A358914, with distinct sums A279785.

For integer partitions we have A382079 (A293511), with distinct sums A382460, (A381870).

With distinct sums we have A382459.

Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360.

Normal multiset partitions: A034691, A035310, A116539, A255906, A381718.

Set systems: A050342, A296120, A318361.

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

Cf. A000110, A000670, A007716, A255903, A275780, A292432, A317532, A326519, 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@@#&&And@@UnsameQ@@@#&]]==1&]], {n,0,5}]