A327350 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity >= k.

1, 1, 0, 2, 1, 0, 9, 5, 2, 0, 114, 84, 44, 17, 0, 6894, 6348, 4983, 3141, 1451, 0, 7785062

Offset: 0

Author

Gus Wiseman, Sep 09 2019

Keywords

• nonn
• tabl
• more

Comments

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.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

If empty edges are allowed, we have T(0,0) = 2.

Examples

			Triangle begins:
     1
     1    0
     2    1    0
     9    5    2    0
   114   84   44   17    0
  6894 6348 4983 3141 1451    0
The antichains counted in row n = 3:
  {123}         {123}         {123}
  {1}{23}       {12}{13}      {12}{13}{23}
  {2}{13}       {12}{23}
  {3}{12}       {13}{23}
  {12}{13}      {12}{13}{23}
  {12}{23}
  {13}{23}
  {1}{2}{3}
  {12}{13}{23}
		

Crossrefs

Column k = 0 is A307249, or A006126 if empty edges are allowed.

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.

Column k = n - 1 is A327020 (cointersecting antichains).

The unlabeled version is A327358.

Negated first differences of rows are A327351.

BII-numbers of antichains are A326704.

Antichain covers are A006126.

Cf. A003465, A014466, A120338, A293606, A293993, A319639, A323818, A327112, A327125, A327334, A327336, A327352, A327356, A327357, 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],Union@@#==Range[n]&&vertConnSys[Range[n],#]>=k&]],{n,0,4},{k,0,n}]

Extensions

a(21) from Robert Price, May 24 2021