cp's OEIS Frontend

A327231 Number of labeled simple connected graphs covering a subset of {1..n} with at least one non-endpoint bridge (non-spanning edge-connectivity 1).

%I A327231 #9 Sep 11 2019 19:41:11
%S A327231 0,0,1,3,18,250,5475,191541,11065572,1104254964,201167132805,
%T A327231 69828691941415,47150542741904118,62354150876493659118,
%U A327231 161919876753750972738791,827272271567137357352991705,8331016130913639432634637862600,165634930763383717802534343776893928
%N A327231 Number of labeled simple connected graphs covering a subset of {1..n} with at least one non-endpoint bridge (non-spanning edge-connectivity 1).
%C A327231 A bridge is an edge whose removal disconnected the graph, while an endpoint is a vertex belonging to only one edge. The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty.
%H A327231 Andrew Howroyd, <a href="/A327231/b327231.txt">Table of n, a(n) for n = 0..50</a>
%F A327231 Binomial transform of A327079.
%e A327231 The a(2) = 1 through a(4) = 18 edge-sets:
%e A327231   {12}  {12}  {12}
%e A327231         {13}  {13}
%e A327231         {23}  {14}
%e A327231               {23}
%e A327231               {24}
%e A327231               {34}
%e A327231               {12,13,24}
%e A327231               {12,13,34}
%e A327231               {12,14,23}
%e A327231               {12,14,34}
%e A327231               {12,23,34}
%e A327231               {12,24,34}
%e A327231               {13,14,23}
%e A327231               {13,14,24}
%e A327231               {13,23,24}
%e A327231               {13,24,34}
%e A327231               {14,23,24}
%e A327231               {14,23,34}
%t A327231 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 A327231 edgeConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
%t A327231 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],edgeConnSys[#]==1&]],{n,0,4}]
%Y A327231 Column k = 1 of A327148.
%Y A327231 The covering version is A327079.
%Y A327231 Connected bridged graphs (spanning edge-connectivity 1) are A327071.
%Y A327231 BII-numbers of set-systems with non-spanning edge-connectivity 1 are A327099.
%Y A327231 Covering set-systems with non-spanning edge-connectivity 1 are A327129.
%Y A327231 Cf. A001187, A052446, A322395, A327072, A327073, A327102.
%K A327231 nonn
%O A327231 0,4
%A A327231 _Gus Wiseman_, Sep 01 2019
%E A327231 Terms a(6) and beyond from _Andrew Howroyd_, Sep 11 2019