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%I A327128 #7 Sep 02 2019 08:04:58
%S A327128 0,1,2,27,2084
%N A327128 Number of set-systems with n vertices whose edge-set has cut-connectivity 1.
%C A327128 A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. We define the cut-connectivity (A326786, A327237, A327126) of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039, A327040), this is the same as vertex-connectivity (A327334, A327051).
%F A327128 Binomial transform of A327197.
%t A327128 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]]]]]]]]];
%t A327128 cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
%t A327128 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],cutConnSys[Union@@#,#]==1&]],{n,0,3}]
%Y A327128 The covering version is A327197.
%Y A327128 The BII-numbers of these set-systems are A327098.
%Y A327128 Cf. A003465, A052442, A052443, A259862, A323818, A326786, A327101, A327112, A327113, A327114, A327126, A327229.
%K A327128 nonn,more
%O A327128 0,3
%A A327128 _Gus Wiseman_, Sep 02 2019