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%I #7 Jun 28 2019 21:14:10
%S 1,0,1,0,1,14,539
%N Number of simple graphs covering the vertices {1..n} whose nesting edges are connected.
%C Covering means there are no isolated vertices. Two edges {a,b}, {c,d} are nesting if a < c < d < b or c < a < b < d. A graph has its nesting edges connected if the graph whose vertices are the edges and whose edges are nesting pairs of edges is connected.
%H Gus Wiseman, <a href="/A326331/a326331.png">The a(5) = 14 simple nesting-connected covering graphs</a>.
%t nesXQ[stn_]:=MatchQ[stn,{___,{x_,y_},___,{z_,t_},___}/;x<z<t<y||z<x<y<t];
%t nestcmpts[stn_]:=csm[Union[List/@stn,Select[Subsets[stn,{2}],nesXQ]]];
%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 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[nestcmpts[#]]<=1&]],{n,0,5}]
%Y The non-covering case is the binomial transform A326330.
%Y Covering graphs whose crossing edges are connected are A324327.
%Y Cf. A006125, A007297, A054726, A099947, A117662, A136653, A324328.
%Y Cf. A326210, A326293, A326335, A326336, A326337, A326338, A326339.
%K nonn,more
%O 0,6
%A _Gus Wiseman_, Jun 27 2019
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