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%I #5 Sep 01 2019 08:40:02
%S 0,1,4,56,4640
%N Number of connected set-systems with n vertices and at least one bridge (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 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.
%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],#]==1&]],{n,0,3}]
%Y The BII-numbers of these set-systems are A327111.
%Y Set systems with non-spanning edge-connectivity 1 are A327196, with covering case A327129.
%Y Set systems with spanning edge-connectivity 2 are A327130.
%Y Cf. A003465, A323818, A327069, A327071, A327099, A327108, A327144.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Aug 27 2019
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