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A327356 Number of connected separable antichains of nonempty sets covering n vertices (vertex-connectivity 1).

%I A327356 #4 Sep 11 2019 20:22:14
%S A327356 0,0,1,3,40,1365
%N A327356 Number of connected separable antichains of nonempty sets covering n vertices (vertex-connectivity 1).
%C A327356 An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.
%C A327356 The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.
%e A327356 Non-isomorphic representatives of the a(4) = 40 set-systems:
%e A327356   {{1,2},{1,3,4}}
%e A327356   {{1,2},{1,3},{1,4}}
%e A327356   {{1,2},{1,3},{2,4}}
%e A327356   {{1,2},{1,3},{1,4},{2,3}}
%t A327356 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A327356 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 A327356 vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]];
%t A327356 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],vertConnSys[Range[n],#]==1&]],{n,0,4}]
%Y A327356 Column k = 1 of A327351.
%Y A327356 The graphical case is A327336.
%Y A327356 The unlabeled version is A327436.
%Y A327356 Cf. A014466, A048143, A307249, A326786, A326950, A327062, A327112, A327114, A327334.
%K A327356 nonn,more
%O A327356 0,4
%A A327356 _Gus Wiseman_, Sep 11 2019