cp's OEIS Frontend

A327113 Number of set-systems covering n vertices with cut-connectivity 2.

%I A327113 #9 Sep 01 2019 22:03:44
%S A327113 0,0,4,0,4752
%N A327113 Number of set-systems covering n vertices with cut-connectivity 2.
%C A327113 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 A327113 The a(2) = 4 set-systems:
%e A327113   {{1,2}}
%e A327113   {{1},{1,2}}
%e A327113   {{2},{1,2}}
%e A327113   {{1},{2},{1,2}}
%t A327113 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 A327113 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 A327113 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&vConn[#]==2&]],{n,0,3}]
%Y A327113 Covering graphs with cut-connectivity >= 2 are A013922, if we assume A013922(2) = 1.
%Y A327113 Covering antichains (blobs) with cut-connectivity >= 2 are A275307, if we assume A275307(1) = 0.
%Y A327113 2-vertex-connected integer partitions are A322387.
%Y A327113 Connected covering set-systems are A323818.
%Y A327113 Covering set-systems with cut-connectivity >= 2 are A327112.
%Y A327113 The cut-connectivity of the set-system with BII-number n is A326786(n).
%Y A327113 BII-numbers of set-systems with cut-connectivity 2 are A327082.
%Y A327113 Cf. A002218, A003465, A048143, A259862, A322389, A327101, A327113, A327126, A327128, A327130.
%K A327113 nonn,more
%O A327113 0,3
%A A327113 _Gus Wiseman_, Aug 24 2019