A322065 - OEIS

%I #6 Nov 26 2018 17:04:26

%S 1,1,1,11,525,146513

%N Number of ways to choose a stable partition of a connected antichain of sets spanning n vertices.

%C A stable partition of a hypergraph or set system is a set partition of the vertices where no non-singleton edge has all its vertices in the same block.

%e The a(3) = 11 stable partitions. The connected antichain is on top, and below is a list of all its stable partitions.

%e {1,2,3}        {1,3}{2,3}     {1,2}{2,3}     {1,2}{1,3}     {1,2}{1,3}{2,3}

%e --------       --------       --------       --------       --------

%e {{1},{2,3}}    {{1,2},{3}}    {{1,3},{2}}    {{1},{2,3}}    {{1},{2},{3}}

%e {{1,2},{3}}    {{1},{2},{3}}  {{1},{2},{3}}  {{1},{2},{3}}

%e {{1,3},{2}}

%e {{1},{2},{3}}

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%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 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[Sum[Length[Select[stableSets[Complement[Subsets[Range[n]],Union@@Subsets/@stn],SubsetQ],And[Union@@#==Range[n],Length[csm[#]]==1]&]],{stn,sps[Range[n]]}],{n,5}]

%Y Cf. A000110, A001187, A006125, A048143, A229048, A240936, A245883, A277203, A321911, A321979, A322063, A322064.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Nov 25 2018