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%I #4 Jun 30 2019 06:51:08
%S 1,1,2,8,46,323,2565
%N Number of non-nesting connected simple graphs on a subset of {1..n}.
%C Two edges {a,b}, {c,d} are nesting if a < c < d < b or c < a < b < d.
%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}]],Length[csm[#]]<=1&&!MatchQ[#,{___,{x_,y_},___,{z_,t_},___}/;x<z<t<y||z<x<y<t]&]],{n,0,5}]
%Y The binomial transform is the covering case A326350.
%Y Connected simple graphs are A001349.
%Y Connected simple graphs with no crossing or nesting edges are A326294.
%Y Simple graphs without crossing or nesting edges are A326244.
%Y Cf. A006125, A054726, A117662, A136653.
%Y Cf. A324169, A326210, A326293, A326329, A326340.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Jun 30 2019