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%I #6 Sep 01 2019 08:42:03
%S 0,1,4,44,2960
%N Number of connected set-systems with n vertices and at least one bridge that is not an endpoint (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 Binomial transform of A327129.
%e Non-isomorphic representatives of the a(3) = 44 set-systems:
%e {{1}}
%e {{1,2}}
%e {{1,2,3}}
%e {{1},{2},{1,2}}
%e {{1},{1,2},{2,3}}
%e {{1},{2},{1,2,3}}
%e {{1},{2,3},{1,2,3}}
%e {{1},{2},{1,2},{1,3}}
%e {{1},{2},{1,3},{2,3}}
%e {{1},{2},{3},{1,2,3}}
%e {{1},{2},{1,3},{1,2,3}}
%e {{1},{2},{3},{1,2},{1,3}}
%e {{1},{2},{3},{1,2},{1,2,3}}
%t 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 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}]],eConn[#]==1&]],{n,0,3}]
%Y The covering version is A327129.
%Y The BII-numbers of these set-systems are A327099.
%Y The restriction to simple graphs is A327231.
%Y Set-systems with spanning edge-connectivity 1 are A327145.
%Y Cf. A263296, A322395, A327079, A327111, A327148, A327149, A327200.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Aug 31 2019
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