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%I #7 Sep 01 2019 08:40:16
%S 0,1,2,35,2804
%N Number of connected set-systems covering n vertices with at least one edge whose removal (along with any non-covered vertices) disconnects the set-system (non-spanning edge-connectivity 1).
%C A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.
%F Inverse binomial transform of A327196.
%e The a(3) = 35 set-systems:
%e {123} {1}{12}{23} {1}{2}{12}{13} {1}{2}{3}{12}{13}
%e {1}{13}{23} {1}{2}{12}{23} {1}{2}{3}{12}{23}
%e {1}{2}{123} {1}{2}{13}{23} {1}{2}{3}{13}{23}
%e {1}{3}{123} {1}{2}{3}{123} {1}{2}{3}{12}{123}
%e {2}{12}{13} {1}{3}{12}{13} {1}{2}{3}{13}{123}
%e {2}{13}{23} {1}{3}{12}{23} {1}{2}{3}{23}{123}
%e {2}{3}{123} {1}{3}{13}{23}
%e {3}{12}{13} {2}{3}{12}{13}
%e {3}{12}{23} {2}{3}{12}{23}
%e {1}{23}{123} {2}{3}{13}{23}
%e {2}{13}{123} {1}{2}{13}{123}
%e {3}{12}{123} {1}{2}{23}{123}
%e {1}{3}{12}{123}
%e {1}{3}{23}{123}
%e {2}{3}{12}{123}
%e {2}{3}{13}{123}
%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 eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&eConn[#]==1&]],{n,0,3}]
%Y The restriction to simple graphs is A327079, with non-covering version A327231.
%Y The version for spanning edge-connectivity is A327145, with BII-numbers A327111.
%Y The BII-numbers of these set-systems are A327099.
%Y The non-covering version is A327196.
%Y Cf. A003465, A006129, A263296, A322395, A323818, A327071, A327149.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Aug 27 2019
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