A320450 Number of strict antichains of sets whose multiset union is an integer partition of n.

1, 1, 1, 3, 3, 5, 10, 13, 19, 28, 47, 64, 98

Offset

0,4

Links

Table of n, a(n) for n=0..12.

Example

The a(1) = 1 through a(8) = 19 antichains:
  {{1}}  {{2}}  {{3}}      {{4}}      {{5}}      {{6}}
                {{1,2}}    {{1,3}}    {{1,4}}    {{1,5}}
                {{1},{2}}  {{1},{3}}  {{2,3}}    {{2,4}}
                                      {{1},{4}}  {{1,2,3}}
                                      {{2},{3}}  {{1},{5}}
                                                 {{2},{4}}
                                                 {{1},{2,3}}
                                                 {{2},{1,3}}
                                                 {{3},{1,2}}
                                                 {{1},{2},{3}}
.
  {{7}}          {{8}}
  {{1,6}}        {{1,7}}
  {{2,5}}        {{2,6}}
  {{3,4}}        {{3,5}}
  {{1,2,4}}      {{1,2,5}}
  {{1},{6}}      {{1,3,4}}
  {{2},{5}}      {{1},{7}}
  {{3},{4}}      {{2},{6}}
  {{1},{2,4}}    {{3},{5}}
  {{2},{1,4}}    {{1},{2,5}}
  {{4},{1,2}}    {{1},{3,4}}
  {{1,2},{1,3}}  {{2},{1,5}}
  {{1},{2},{4}}  {{3},{1,4}}
                 {{4},{1,3}}
                 {{5},{1,2}}
                 {{1,2},{1,4}}
                 {{1,2},{2,3}}
                 {{1},{2},{5}}
                 {{1},{3},{4}}

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]]]];
submultisetQ[M_, N_]:=Or[Length[M]==0, MatchQ[{Sort[List@@M], Sort[List@@N]}, {{x_, Z___}, {___, x_, W___}}/; submultisetQ[{Z}, {W}]]];
antiQ[s_]:=Select[Tuples[s, 2], And[UnsameQ@@#, submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n], And[UnsameQ@@#, And@@UnsameQ@@@#, antiQ[#]]&]], {n, 10}]

Crossrefs

Cf. A001970, A050342, A089259, A258466, A261049, A319719, A319721, A320328, A320331, A320353.

Sequence in context: A331925 A252943 A294617 * A100886 A326175 A072337

Adjacent sequences: A320447 A320448 A320449 * A320451 A320452 A320453

Keyword

nonn,more

Author

Gus Wiseman, Oct 12 2018

Status

approved