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

1, 1, 3, 11, 51, 259, 1461, 9133

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..7.

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

Example

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}}

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}]

Crossrefs

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

Sequence in context: A359146 A199212 A113283 * A127784 A112696 A346967

Adjacent sequences: A326516 A326517 A326518 * A326520 A326521 A326522

Keyword

nonn,more

Author

Gus Wiseman, Jul 12 2019

Status

approved