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%I #5 Sep 01 2019 22:03:59
%S 0,1,0,24,1984
%N Number of set-systems covering n vertices with cut-connectivity 1.
%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 resulting empty edges) to obtain in a disconnected or empty set-system. Except for cointersecting set-systems (A327040), this is the same as vertex-connectivity.
%F Inverse binomial transform of A327128.
%e The a(3) = 24 set-systems:
%e {12}{13} {1}{12}{13} {1}{2}{12}{13} {1}{2}{3}{12}{13}
%e {12}{23} {1}{12}{23} {1}{2}{12}{23} {1}{2}{3}{12}{23}
%e {13}{23} {1}{13}{23} {1}{2}{13}{23} {1}{2}{3}{13}{23}
%e {2}{12}{13} {1}{3}{12}{13}
%e {2}{12}{23} {1}{3}{12}{23}
%e {2}{13}{23} {1}{3}{13}{23}
%e {3}{12}{13} {2}{3}{12}{13}
%e {3}{12}{23} {2}{3}{12}{23}
%e {3}{13}{23} {2}{3}{13}{23}
%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 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 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&cutConnSys[Range[n],#]==1&]],{n,0,3}]
%Y The BII-numbers of these set-systems are A327098.
%Y The same for cut-connectivity 2 is A327113.
%Y The non-covering version is A327128.
%Y Cf. A003465, A052442, A052443, A259862, A323818, A326786, A327101, A327112, A327114, A327126, A327229.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Sep 01 2019
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