A326519 Number of normal multiset partitions of weight n where each part has a different sum.

1, 1, 3, 11, 51, 259, 1461, 9133, 62348, 459547, 3632419

Offset: 0

Author

Gus Wiseman, Jul 12 2019

Keywords

• nonn
• more

Comments

A multiset partition is normal if it covers an initial interval of positive integers.

Examples

			The a(0) = 1 through a(3) = 11 normal multiset partitions:
  {}  {{1}}  {{1,1}}    {{1,1,1}}
             {{1,2}}    {{1,1,2}}
             {{1},{2}}  {{1,2,2}}
                        {{1,2,3}}
                        {{1},{1,1}}
                        {{1},{1,2}}
                        {{1},{2,2}}
                        {{1},{2,3}}
                        {{2},{1,2}}
                        {{2},{1,3}}
                        {{1},{2},{3}}
		

Links

• Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Crossrefs

Cf. A038041, A255906, A275780, A317583, A321469, A326517, A326518, A326520, A326521, A326535.

Programs

Mathematica

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Total/@#&]],{n,0,5}]

Extensions

a(8)-a(10) from Robert Price, Apr 03 2025