A327113 - OEIS

%I #9 Sep 01 2019 22:03:44

%S 0,0,4,0,4752

%N Number of set-systems covering n vertices with cut-connectivity 2.

%C 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).

%e The a(2) = 4 set-systems:

%e   {{1,2}}

%e   {{1},{1,2}}

%e   {{2},{1,2}}

%e   {{1},{2},{1,2}}

%t 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]]]]]]]]];

%t 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]]];

%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&vConn[#]==2&]],{n,0,3}]

%Y Covering graphs with cut-connectivity >= 2 are A013922, if we assume A013922(2) = 1.

%Y Covering antichains (blobs) with cut-connectivity >= 2 are A275307, if we assume A275307(1) = 0.

%Y 2-vertex-connected integer partitions are A322387.

%Y Connected covering set-systems are A323818.

%Y Covering set-systems with cut-connectivity >= 2 are A327112.

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

%Y BII-numbers of set-systems with cut-connectivity 2 are A327082.

%Y Cf. A002218, A003465, A048143, A259862, A322389, A327101, A327113, A327126, A327128, A327130.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Aug 24 2019