A326337 Number of simple graphs covering the vertices {1..n} whose weakly nesting edges are connected.

1, 0, 1, 3, 29, 595, 23437

Offset: 0

Author

Gus Wiseman, Jun 28 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) = 29 covering simple graphs whose weakly nesting edges are connected.

Crossrefs

The binomial transform is the non-covering case A326338.

The non-weak case is A326331.

Simple graphs whose nesting edges are connected are A326330.

Cf. A006125, A007297, A054726, A099947, A136653.

Cf. A324169, A324327, A326293, A326294, A326329, A326335, A326336.

Programs

Mathematica

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