A382430 Number of non-isomorphic finite multisets of size n that cannot be partitioned into sets with distinct sums.

0, 0, 1, 1, 2, 3, 5, 6, 9, 12, 17, 22, 32

Offset: 0

Author

Gus Wiseman, Apr 01 2025

Keywords

• nonn
• more

Comments

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

			The a(2) = 1 through a(7) = 6 multisets:
  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}  {1,1,1,1,1,1}  {1,1,1,1,1,1,1}
                  {1,1,1,2}  {1,1,1,1,2}  {1,1,1,1,1,2}  {1,1,1,1,1,1,2}
                             {1,1,1,2,2}  {1,1,1,1,2,2}  {1,1,1,1,1,2,2}
                                          {1,1,1,1,2,3}  {1,1,1,1,1,2,3}
                                          {1,1,1,2,2,2}  {1,1,1,1,2,2,2}
                                                         {1,1,1,1,2,2,3}
		

Crossrefs

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

The strict version is A292444.

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 A381990, ranks A381806, complement A381992, ranks A382075.

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

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

The complement is counted by A382523, strict A381996.

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}]