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%I #6 Aug 29 2018 16:49:13
%S 2,4,18,450,436270
%N Number of finite sets of set partitions of {1,...,n} such that any two have join {{1,...,n}}.
%e The a(3) = 18 sets of set partitions:
%e 0
%e {{1,2,3}}
%e {{1,3},{2}}
%e {{1,2},{3}}
%e {{1},{2,3}}
%e {{1},{2},{3}}
%e {{1,3},{2}} {{1,2,3}}
%e {{1,2},{3}} {{1,2,3}}
%e {{1,2},{3}} {{1,3},{2}}
%e {{1},{2,3}} {{1,2,3}}
%e {{1},{2,3}} {{1,3},{2}}
%e {{1},{2,3}} {{1,2},{3}}
%e {{1},{2},{3}} {{1,2,3}}
%e {{1,2},{3}} {{1,3},{2}} {{1,2,3}}
%e {{1},{2,3}} {{1,3},{2}} {{1,2,3}}
%e {{1},{2,3}} {{1,2},{3}} {{1,2,3}}
%e {{1},{2,3}} {{1,2},{3}} {{1,3},{2}}
%e {{1},{2,3}} {{1,2},{3}} {{1,3},{2}} {{1,2,3}}
%t stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t 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 Table[Length[stableSets[sps[Range[n]],Length[csm[Union[#1,#2]]]>1&]],{n,4}]
%Y Cf. A000110, A000258, A008277, A059849, A060639, A096827, A181939.
%Y Cf. A318389, A318390, A318391, A318392, A318394, A318532.
%K nonn,more
%O 1,1
%A _Gus Wiseman_, Aug 28 2018
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