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A327130 Number of set-systems covering n vertices with spanning edge-connectivity 2.

%I A327130 #5 Sep 01 2019 08:40:09
%S A327130 0,0,0,32,9552
%N A327130 Number of set-systems covering n vertices with spanning edge-connectivity 2.
%C A327130 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 A327130 The a(3) = 32 set-systems:
%e A327130 {12}{13}{23}  {1}{12}{13}{23}  {1}{2}{12}{13}{23}  {1}{2}{3}{12}{13}{23}
%e A327130 {12}{13}{123} {2}{12}{13}{23}  {1}{3}{12}{13}{23}  {1}{2}{3}{12}{13}{123}
%e A327130 {12}{23}{123} {3}{12}{13}{23}  {2}{3}{12}{13}{23}  {1}{2}{3}{12}{23}{123}
%e A327130 {13}{23}{123} {1}{12}{13}{123} {1}{2}{12}{13}{123} {1}{2}{3}{13}{23}{123}
%e A327130               {1}{12}{23}{123} {1}{2}{12}{23}{123}
%e A327130               {1}{13}{23}{123} {1}{2}{13}{23}{123}
%e A327130               {2}{12}{13}{123} {1}{3}{12}{13}{123}
%e A327130               {2}{12}{23}{123} {1}{3}{12}{23}{123}
%e A327130               {2}{13}{23}{123} {1}{3}{13}{23}{123}
%e A327130               {3}{12}{13}{123} {2}{3}{12}{13}{123}
%e A327130               {3}{12}{23}{123} {2}{3}{12}{23}{123}
%e A327130               {3}{13}{23}{123} {2}{3}{13}{23}{123}
%t A327130 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 A327130 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
%t A327130 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],spanEdgeConn[Range[n],#]==2&]],{n,0,3}]
%Y A327130 The BII-numbers of these set-systems are A327108.
%Y A327130 Set-systems with spanning edge-connectivity 1 are A327145.
%Y A327130 The restriction to simple graphs is A327146.
%Y A327130 Cf. A003465, A323818, A327069, A327109, A327111, A327144.
%K A327130 nonn,more
%O A327130 0,4
%A A327130 _Gus Wiseman_, Aug 27 2019