A320353 Number of antichains of multisets whose multiset union is an integer partition of n.

1, 1, 3, 5, 11, 17, 36, 56, 107, 175, 311, 505, 887

Offset: 0

Author

Gus Wiseman, Oct 11 2018

Keywords

• nonn
• more

Examples

			The a(1) = 1 through a(5) = 17 antichains:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{2}}      {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{1},{3}}          {{1,2,2}}
                                   {{2},{2}}          {{1},{4}}
                                   {{1,1,1,1}}        {{2},{3}}
                                   {{2},{1,1}}        {{1,1,1,2}}
                                   {{1,1},{1,1}}      {{1},{2,2}}
                                   {{1},{1},{2}}      {{3},{1,1}}
                                   {{1},{1},{1},{1}}  {{1,1,1,1,1}}
                                                      {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{2},{1,1,1}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}
		

Links

• Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Crossrefs

Cf. A001970, A006126, A050342, A089259, A258466, A261005, A261049, A293994, A319719, A319721, A320328, A320355, A320356.

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],antiQ]],{n,8}]