A327436 -id:A327436 - cp's OEIS frontend

A327359 Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.

1, 1, 0, 1, 1, 0, 2, 1, 2, 0, 6, 4, 4, 6, 0, 23, 29, 37, 37, 54, 0

Offset: 0

Gus Wiseman, Sep 10 2019

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.

			Triangle begins:
   1
   1  0
   1  1  0
   2  1  2  0
   6  4  4  6  0
  23 29 37 37 54  0
Row n = 4 counts the following antichains:
{1}{234}      {14}{234}        {134}{234}           {1234}
{12}{34}      {13}{24}{34}     {13}{14}{234}        {12}{134}{234}
{1}{2}{34}    {14}{24}{34}     {12}{13}{24}{34}     {124}{134}{234}
{1}{24}{34}   {14}{23}{24}{34} {13}{14}{23}{24}{34} {12}{13}{14}{234}
{1}{2}{3}{4}                                        {123}{124}{134}{234}
{1}{23}{24}{34}                                     {12}{13}{14}{23}{24}{34}

Row sums are A261005, or A006602 if empty edges are allowed.
Column k = 0 is A327426.
Column k = 1 is A327436.
Column k = n - 1 is A327425.
The labeled version is A327351.
Cf. A003465, A006126, A014466, A048143, A293993, A323818, A326704, A327125, A327334, A327336, A327350, A327358.

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

Original entry on oeis.org

0, 0, 1, 3, 40, 1365

Offset: 0

Gus Wiseman, Sep 11 2019

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 resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

			Non-isomorphic representatives of the a(4) = 40 set-systems:
  {{1,2},{1,3,4}}
  {{1,2},{1,3},{1,4}}
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3}}

Column k = 1 of A327351.
The graphical case is A327336.
The unlabeled version is A327436.
Cf. A014466, A048143, A307249, A326786, A326950, A327062, A327112, A327114, A327334.

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],#]==1&]],{n,0,4}]

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A327359 Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.

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