A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

0, 0, 1, 1, 3, 5, 9, 16, 27, 48, 78, 133

Offset: 0

Author

Gus Wiseman, Mar 29 2025

Keywords

• nonn
• more

Comments

First differs from A292432 at a(9) = 48, A292432(9) = 46.

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 m = {1,1,1,2,2} has 3 partitions into a set of sets:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
but none of these has distinct block-sums, so m is counted under a(5).
The a(2) = 1 through a(6) = 9 normal multisets:
  {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}
                  {1,2,2,2}  {1,1,1,2,2}  {1,1,1,1,2,2}
                             {1,1,2,2,2}  {1,1,1,1,2,3}
                             {1,2,2,2,2}  {1,1,1,2,2,2}
                                          {1,1,2,2,2,2}
                                          {1,2,2,2,2,2}
                                          {1,2,2,2,2,3}
                                          {1,2,3,3,3,3}
		

Crossrefs

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

Without distinct sums we have A292432, complement A382214.

The strongly normal version without distinct sums is A292444, complement A381996.

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

Normal multiset partitions of this type are counted by A381718, without distinct sums A116539.

For integer partitions the complement is A381990, ranks A381806, without distinct sums A382078, ranks A293243.

For integer partitions we have A381992, ranks A382075, without distinct sums A382077, ranks A382200.

The complement is counted by A382216.

The strongly normal version is A382430, complement A382460.

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

A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.

A001055 count factorizations, strict A045778.

Normal multiset partitions: A034691, A035310, A255906.

Set systems: A050342, A296120, A318361.

Set multipartitions: A089259, A270995, A296119, A318360.

Cf. A000110, A007716, A050320, A116540, A255903, A275780, A321469, A326517, A326518, A326519, A382428, A382429.

Programs

Mathematica

allnorm[n_Integer]:=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}]