A326349 -id:A326349 - cp's OEIS frontend

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

1, 1, 2, 4, 8, 27, 192, 1750

Offset: 0

Gus Wiseman, Jun 29 2019

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.

Gus Wiseman, The a(5) = 27 non-nesting, topologically connected simple graphs.

The inverse binomial transform is the covering case A326349.
Topologically connected simple graphs are A324328.
Non-crossing simple graphs are A054726.
Topologically connected set partitions are A099947.
Cf. A000108, A000699, A006125, A007297, A117662, A136653.
Cf. A324173, A324323, A324327, A326244, A326330, A326341.

croXQ[eds_]:=MatchQ[eds,{_,{x_,y_},_,{z_,t_},_}/;x_,{x_,y_},_,{z_,t_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],!nesXQ[#]&&Length[csm[Union[Subsets[#,{1}],Select[Subsets[#,{2}],croXQ]]]]<=1&]],{n,0,5}]

cp's OEIS Frontend

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

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