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

1, 1, 2, 7, 48, 781, 27518

Offset: 0

Author

Gus Wiseman, Jun 29 2019

Keywords

• nonn
• more

Comments

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.

Links

• Gus Wiseman, The a(4) = 48 weakly nesting-connected graphs.

Crossrefs

The inverse binomial transform is the covering case A326337.

The non-weak case is A326330.

Cf. A006125, A099947, A136653, A324328, A326289, A326293, A326331, A326335, A326336, A326341.

Programs

Mathematica

wknXQ[eds_]:=MatchQ[eds,{_,{x_,y_},_,{z_,t_},_}/;(x<=z&&y>=t)||(x>=z&&y<=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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[Union[List/@#,Select[Subsets[#,{2}],wknXQ]]]]<=1&]],{n,0,5}]