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

1, 2, 0, 5, 1, 0, 19, 5, 2, 0, 167, 84, 44, 17, 0

Offset: 0

Author

Gus Wiseman, Sep 26 2019

Keywords

• nonn
• more
• tabl

Comments

An antichain is a set of nonempty sets, none of which is a subset of any other.

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.

Examples

			Triangle begins:
    1
    2   0
    5   1   0
   19   5   2   0
  167  84  44  17   0
		

Crossrefs

Except for the first column, same as the covering case A327350.

Column k = 0 is A014466 (antichains).

Column k = 1 is A048143 (clutters), if we assume A048143(0) = A048143(1) = 0.

Column k = 2 is A275307 (blobs), if we assume A275307(1) = A275307(2) = 0.

The unlabeled version is A327807.

The case for vertex connectivity exactly k is A327351.

Cf. A014466, A293606, A326704, A327057, A327062, A327125, A327358.

Programs

Mathematica

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]]]]]]]]];
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]]]];
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]}]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],vertConnSys[Range[n],#]>=k&]],{n,0,4},{k,0,n}]