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A327806 Triangle read by rows where T(n,k) is the number of antichains of sets with n vertices and vertex-connectivity >= k.

%I A327806 #4 Sep 26 2019 21:15:00
%S A327806 1,2,0,5,1,0,19,5,2,0,167,84,44,17,0
%N A327806 Triangle read by rows where T(n,k) is the number of antichains of sets with n vertices and vertex-connectivity >= k.
%C A327806 An antichain is a set of nonempty sets, none of which is a subset of any other.
%C A327806 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 A327806 Triangle begins:
%e A327806     1
%e A327806     2   0
%e A327806     5   1   0
%e A327806    19   5   2   0
%e A327806   167  84  44  17   0
%t A327806 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 A327806 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 A327806 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 A327806 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],vertConnSys[Range[n],#]>=k&]],{n,0,4},{k,0,n}]
%Y A327806 Except for the first column, same as the covering case A327350.
%Y A327806 Column k = 0 is A014466 (antichains).
%Y A327806 Column k = 1 is A048143 (clutters), if we assume A048143(0) = A048143(1) = 0.
%Y A327806 Column k = 2 is A275307 (blobs), if we assume A275307(1) = A275307(2) = 0.
%Y A327806 The unlabeled version is A327807.
%Y A327806 The case for vertex connectivity exactly k is A327351.
%Y A327806 Cf. A014466, A293606, A326704, A327057, A327062, A327125, A327358.
%K A327806 nonn,more,tabl
%O A327806 0,2
%A A327806 _Gus Wiseman_, Sep 26 2019