A319255 Number of strict antichains of multisets whose multiset union is an integer partition of n.

1, 1, 2, 4, 7, 13, 23, 41, 70, 123, 208, 355, 597

Offset: 0

Author

Gus Wiseman, Oct 12 2018

Keywords

• nonn
• more

Examples

			The a(1) = 1 through a(6) = 23 antichains:
  {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
         {{1,1}}  {{1,2}}    {{1,3}}      {{1,4}}        {{1,5}}
                  {{1,1,1}}  {{2,2}}      {{2,3}}        {{2,4}}
                  {{1},{2}}  {{1,1,2}}    {{1,1,3}}      {{3,3}}
                             {{1},{3}}    {{1,2,2}}      {{1,1,4}}
                             {{1,1,1,1}}  {{1},{4}}      {{1,2,3}}
                             {{2},{1,1}}  {{2},{3}}      {{1},{5}}
                                          {{1,1,1,2}}    {{2,2,2}}
                                          {{1},{2,2}}    {{2},{4}}
                                          {{3},{1,1}}    {{1,1,1,3}}
                                          {{1,1,1,1,1}}  {{1,1,2,2}}
                                          {{1,1},{1,2}}  {{1},{2,3}}
                                          {{2},{1,1,1}}  {{2},{1,3}}
                                                         {{3},{1,2}}
                                                         {{4},{1,1}}
                                                         {{1,1,1,1,2}}
                                                         {{1,1},{1,3}}
                                                         {{1,1},{2,2}}
                                                         {{1},{2},{3}}
                                                         {{3},{1,1,1}}
                                                         {{1,1,1,1,1,1}}
                                                         {{1,2},{1,1,1}}
                                                         {{2},{1,1,1,1}}
		

Links

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

Crossrefs

Cf. A001970, A258466, A261049, A319719, A319721, A320353, 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],And[UnsameQ@@#,antiQ[#]]&]],{n,10}]