A381996 Number of non-isomorphic multisets of size n that can be partitioned into a set of sets.

1, 1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 34, 47

Offset: 0

Author

Gus Wiseman, Mar 31 2025

Keywords

• nonn
• more

Comments

First differs from A382523 at a(12) = 47, A382523(12) = 45.

We call a multiset non-isomorphic iff it covers an initial interval of positive integers with weakly decreasing multiplicities. The size of a multiset is the number of elements, counting multiplicity.

Examples

			Differs from A382523 in counting the following under a(12):
  {1,1,1,1,1,1,2,2,3,3,4,5} with partition {{1},{1,2},{1,3},{1,4},{1,5},{1,2,3}}
  {1,1,1,1,2,2,2,2,3,3,3,3} with partition {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
		

Crossrefs

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

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

The complement is counted by A292444.

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

For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.

Weak version is A382214, complement A292432, distinct sums A382216, complement A382202.

For distinct sums we have A382523, complement A382430.

Normal multiset partitions: A034691, A035310, A116540, A255906.

Set systems: A050342, A296120, A318361.

Set multipartitions: A089259, A270995, A296119, A318360.

Cf. A000110, A000670, A007716, A050320, A255903, A317532, A326519, A381992, A382428, A382458, A382459.

Programs

Mathematica

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n];
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[strnorm[n], Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]!={}&]], {n,0,5}]