%I #5 Sep 01 2019 08:40:09
%S 0,0,0,32,9552
%N Number of set-systems covering n vertices with spanning edge-connectivity 2.
%C A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.
%e The a(3) = 32 set-systems:
%e {12}{13}{23} {1}{12}{13}{23} {1}{2}{12}{13}{23} {1}{2}{3}{12}{13}{23}
%e {12}{13}{123} {2}{12}{13}{23} {1}{3}{12}{13}{23} {1}{2}{3}{12}{13}{123}
%e {12}{23}{123} {3}{12}{13}{23} {2}{3}{12}{13}{23} {1}{2}{3}{12}{23}{123}
%e {13}{23}{123} {1}{12}{13}{123} {1}{2}{12}{13}{123} {1}{2}{3}{13}{23}{123}
%e {1}{12}{23}{123} {1}{2}{12}{23}{123}
%e {1}{13}{23}{123} {1}{2}{13}{23}{123}
%e {2}{12}{13}{123} {1}{3}{12}{13}{123}
%e {2}{12}{23}{123} {1}{3}{12}{23}{123}
%e {2}{13}{23}{123} {1}{3}{13}{23}{123}
%e {3}{12}{13}{123} {2}{3}{12}{13}{123}
%e {3}{12}{23}{123} {2}{3}{12}{23}{123}
%e {3}{13}{23}{123} {2}{3}{13}{23}{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 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],spanEdgeConn[Range[n],#]==2&]],{n,0,3}]
%Y The BII-numbers of these set-systems are A327108.
%Y Set-systems with spanning edge-connectivity 1 are A327145.
%Y The restriction to simple graphs is A327146.
%Y Cf. A003465, A323818, A327069, A327109, A327111, A327144.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Aug 27 2019