A327112 Number of set-systems covering n vertices with cut-connectivity >= 2, or 2-cut-connected set-systems.

0, 0, 4, 72, 29856

Offset: 0

Author

Gus Wiseman, Aug 24 2019

Keywords

• nonn
• more

Comments

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The cut-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 disconnected or empty set-system. Except for cointersecting set-systems (A327040), this is the same as vertex-connectivity (A327334, A327051).

Examples

			Non-isomorphic representatives of the a(3) = 72 set-systems:
  {{123}}
  {{3}{123}}
  {{23}{123}}
  {{2}{3}{123}}
  {{1}{23}{123}}
  {{3}{23}{123}}
  {{12}{13}{23}}
  {{13}{23}{123}}
  {{1}{2}{3}{123}}
  {{1}{3}{23}{123}}
  {{2}{3}{23}{123}}
  {{3}{12}{13}{23}}
  {{2}{13}{23}{123}}
  {{3}{13}{23}{123}}
  {{12}{13}{23}{123}}
  {{1}{2}{3}{23}{123}}
  {{2}{3}{12}{13}{23}}
  {{1}{2}{13}{23}{123}}
  {{2}{3}{13}{23}{123}}
  {{3}{12}{13}{23}{123}}
  {{1}{2}{3}{12}{13}{23}}
  {{1}{2}{3}{13}{23}{123}}
  {{2}{3}{12}{13}{23}{123}}
  {{1}{2}{3}{12}{13}{23}{123}}
		

Crossrefs

Covering 2-cut-connected graphs are A013922, if we assume A013922(2) = 1.

Covering 1-cut-connected antichains (clutters) are A048143, if we assume A048143(0) = A048143(1) =0.

Covering 2-cut-connected antichains (blobs) are A275307, if we assume A275307(1) = 0.

Covering set-systems with cut-connectivity 2 are A327113.

2-vertex-connected integer partitions are A322387.

BII-numbers of set-systems with cut-connectivity >= 2 are A327101.

The cut-connectivity of the set-system with BII-number n is A326786(n).

Cf. A002218, A003465, A013922, A259862, A323818, A327082, A327126, A327130.

Programs

Mathematica

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vConn[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&vConn[#]>=2&]],{n,0,3}]