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%I #13 Sep 11 2019 20:21:40
%S 0,0,1,0,12,180,4200,157920,9673664,1011129840,190600639200,
%T 67674822473280,46325637863907072,61746583700640860736,
%U 161051184122415878112640,824849999242893693424992000,8317799170120961768715123118080
%N Number of labeled simple connected graphs covering n vertices with at least one bridge that is not an endpoint/leaf (non-spanning edge-connectivity 1).
%C A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Graphs with no bridges are counted by A095983 (2-edge-connected graphs).
%C Also labeled simple connected graphs covering n vertices with non-spanning edge-connectivity 1, where the non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty graph.
%F a(n) = A001187(n) - A322395(n) for n > 2. - _Andrew Howroyd_, Aug 27 2019
%F Inverse binomial transform of A327231.
%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],{2}]],Union@@#==Range[n]&&eConn[#]==1&]],{n,0,4}]
%Y Column k = 1 of A327149.
%Y The non-covering version is A327231.
%Y Connected bridged graphs (spanning edge-connectivity 1) are A327071.
%Y BII-numbers of graphs with non-spanning edge-connectivity 1 are A327099.
%Y Covering set-systems with non-spanning edge-connectivity 1 are A327129.
%Y Cf. A001187, A006129, A052446, A059166, A322395, A327072, A327073, A327148.
%K nonn
%O 0,5
%A _Gus Wiseman_, Aug 25 2019
%E Terms a(6) and beyond from _Andrew Howroyd_, Aug 27 2019
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