A382523 Number of non-isomorphic finite multisets of size n that can be partitioned into sets with distinct sums.

1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 34, 45

Offset: 0

Author

Gus Wiseman, Apr 01 2025

Keywords

• nonn
• more

Comments

First differs from A381996 at a(12) = 45, A381996(12) = 47.

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

			First differs from A381996 in not counting the following under a(12):
  {1,1,1,1,1,1,2,2,3,3,4,5}
  {1,1,1,1,2,2,2,2,3,3,3,3}
The a(1) = 1 through a(6) = 6 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  {1,1,1,2,2,3}
              {1,2,3}  {1,1,2,3}  {1,1,2,2,3}  {1,1,1,2,3,4}
                       {1,2,3,4}  {1,1,2,3,4}  {1,1,2,2,3,3}
                                  {1,2,3,4,5}  {1,1,2,2,3,4}
                                               {1,1,2,3,4,5}
                                               {1,2,3,4,5,6}
		

Crossrefs

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

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

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

For integer partitions we have A381992, ranks A382075, complement A381990, ranks A381806.

The strict version is A381996.

The strict version for integer partitions is A382077, ranks A382200, complement A382078, ranks A293243.

The labeled version is A382216, complement A382202, strict A382214, complement A292432.

The complement is counted by A382430, strict A292444.

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@@Total/@#&&And@@UnsameQ@@@#&]!={}&]],{n,0,5}]