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

Author

Gus Wiseman, Oct 12 2018

Keywords

• nonn
• more

Examples

			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}}
		

Crossrefs

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

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]]]];
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}]