cp's OEIS Frontend

A326293 Number of non-nesting, topologically connected simple graphs with vertices {1..n}.

%I A326293 #6 Jun 30 2019 06:50:14
%S A326293 1,1,2,4,8,27,192,1750
%N A326293 Number of non-nesting, topologically connected simple graphs with vertices {1..n}.
%C A326293 Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d. A graph with positive integer vertices is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected.
%H A326293 Gus Wiseman, <a href="/A326293/a326293.png">The a(5) = 27 non-nesting, topologically connected simple graphs</a>.
%t A326293 croXQ[eds_]:=MatchQ[eds,{___,{x_,y_},___,{z_,t_},___}/;x<z<y<t||z<x<t<y];
%t A326293 nesXQ[eds_]:=MatchQ[eds,{___,{x_,y_},___,{z_,t_},___}/;x<z<t<y||z<x<y<t];
%t A326293 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 A326293 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],!nesXQ[#]&&Length[csm[Union[Subsets[#,{1}],Select[Subsets[#,{2}],croXQ]]]]<=1&]],{n,0,5}]
%Y A326293 The inverse binomial transform is the covering case A326349.
%Y A326293 Topologically connected simple graphs are A324328.
%Y A326293 Non-crossing simple graphs are A054726.
%Y A326293 Topologically connected set partitions are A099947.
%Y A326293 Cf. A000108, A000699, A006125, A007297, A117662, A136653.
%Y A326293 Cf. A324173, A324323, A324327, A326244, A326330, A326341.
%K A326293 nonn,more
%O A326293 0,3
%A A326293 _Gus Wiseman_, Jun 29 2019