A326520 Number of normal multiset partitions of weight n where every part has the same average.

1, 1, 3, 7, 17, 35, 103, 197

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(4) = 17 normal multiset partitions where every part has the same average:
  {}  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
             {{1,2}}    {{1,1,2}}      {{1,1,1,2}}
             {{1},{1}}  {{1,2,2}}      {{1,1,2,2}}
                        {{1,2,3}}      {{1,1,2,3}}
                        {{1},{1,1}}    {{1,2,2,2}}
                        {{2},{1,3}}    {{1,2,2,3}}
                        {{1},{1},{1}}  {{1,2,3,3}}
                                       {{1,2,3,4}}
                                       {{1},{1,1,1}}
                                       {{1,1},{1,1}}
                                       {{1,2},{1,2}}
                                       {{1,3},{2,2}}
                                       {{1,4},{2,3}}
                                       {{2},{1,2,3}}
                                       {{1},{1},{1,1}}
                                       {{2},{2},{1,3}}
                                       {{1},{1},{1},{1}}

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], SameQ@@Mean/@#&]], {n, 0, 5}]

Crossrefs

Cf. A038041, A051293, A255906, A317583, A326512, A326515, A326517, A326518, A326519, A326521, A326536.

Sequence in context: A114100 A260677 A014395 * A295146 A014314 A221792

Adjacent sequences: A326517 A326518 A326519 * A326521 A326522 A326523

Keyword

nonn,more

Author

Gus Wiseman, Jul 12 2019

Status

approved