OFFSET
0,4
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 resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.
EXAMPLE
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}}
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], #]==1&]], {n, 0, 4}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 11 2019
STATUS
approved