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

0,2

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.

Links

Table of n, a(n) for n=0..14.

Example

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

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}]

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.

Sequence in context: A177267 A188445 A350158 * A319683 A294137 A246723

Adjacent sequences: A327803 A327804 A327805 * A327807 A327808 A327809

Keyword

nonn,more,tabl

Author

Gus Wiseman, Sep 26 2019

Status

approved