cp's OEIS Frontend

A318403 Number of strict connected antichains of sets whose multiset union is an integer partition of n.

%I A318403 #11 Oct 12 2018 22:42:46
%S A318403 1,1,1,2,2,3,4,6,8,12,13,22,31
%N A318403 Number of strict connected antichains of sets whose multiset union is an integer partition of n.
%e A318403 The a(1) = 1 through a(10) = 13 clutters:
%e A318403   {{1}}  {{2}}  {{3}}    {{4}}    {{5}}    {{6}}      {{7}}
%e A318403                 {{1,2}}  {{1,3}}  {{1,4}}  {{1,5}}    {{1,6}}
%e A318403                                   {{2,3}}  {{2,4}}    {{2,5}}
%e A318403                                            {{1,2,3}}  {{3,4}}
%e A318403                                                       {{1,2,4}}
%e A318403                                                       {{1,2},{1,3}}
%e A318403 .
%e A318403   {{8}}          {{9}}          {{10}}
%e A318403   {{1,7}}        {{1,8}}        {{1,9}}
%e A318403   {{2,6}}        {{2,7}}        {{2,8}}
%e A318403   {{3,5}}        {{3,6}}        {{3,7}}
%e A318403   {{1,2,5}}      {{4,5}}        {{4,6}}
%e A318403   {{1,3,4}}      {{1,2,6}}      {{1,2,7}}
%e A318403   {{1,2},{1,4}}  {{1,3,5}}      {{1,3,6}}
%e A318403   {{1,2},{2,3}}  {{2,3,4}}      {{1,4,5}}
%e A318403                  {{1,2},{1,5}}  {{2,3,5}}
%e A318403                  {{1,2},{2,4}}  {{1,2,3,4}}
%e A318403                  {{1,3},{1,4}}  {{1,2},{1,6}}
%e A318403                  {{1,3},{2,3}}  {{1,2},{2,5}}
%e A318403                                 {{1,3},{1,5}}
%t A318403 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A318403 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A318403 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A318403 submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{___,x_,W___}}/;submultisetQ[{Z},{W}]]];
%t A318403 antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
%t A318403 Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,And@@UnsameQ@@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,8}]
%Y A318403 Cf. A001970, A007718, A048143, A050342, A056156, A089259, A261049, A293994, A319719, A320351, A320353, A320355, A320356.
%K A318403 nonn,more
%O A318403 0,4
%A A318403 _Gus Wiseman_, Oct 12 2018