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A326338 Number of simple graphs with vertices {1..n} whose weakly nesting edges are connected.

%I A326338 #6 Jun 30 2019 06:50:22
%S A326338 1,1,2,7,48,781,27518
%N A326338 Number of simple graphs with vertices {1..n} whose weakly nesting edges are connected.
%C A326338 Two edges {a,b}, {c,d} are weakly nesting if a <= c < d <= b or c <= a < b <= d. A graph has its weakly nesting edges connected if the graph whose vertices are the edges and whose edges are weakly nesting pairs of edges is connected.
%H A326338 Gus Wiseman, <a href="/A326338/a326338.png">The a(4) = 48 weakly nesting-connected graphs</a>.
%t A326338 wknXQ[eds_]:=MatchQ[eds,{___,{x_,y_},___,{z_,t_},___}/;(x<=z&&y>=t)||(x>=z&&y<=t)];
%t A326338 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 A326338 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[Union[List/@#,Select[Subsets[#,{2}],wknXQ]]]]<=1&]],{n,0,5}]
%Y A326338 The inverse binomial transform is the covering case A326337.
%Y A326338 The non-weak case is A326330.
%Y A326338 Cf. A006125, A099947, A136653, A324328, A326289, A326293, A326331, A326335, A326336, A326341.
%K A326338 nonn,more
%O A326338 0,3
%A A326338 _Gus Wiseman_, Jun 29 2019