A326337 - OEIS

%I #4 Jun 28 2019 21:14:55

%S 1,0,1,3,29,595,23437

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

%C 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.

%H Gus Wiseman, <a href="/A326337/a326337.png">The a(4) = 29 covering simple graphs whose weakly nesting edges are connected</a>.

%t wknXQ[stn_]:=MatchQ[stn,{___,{___,x_,y_,___},___,{___,z_,t_,___},___}/;(x<=z&&y>=t)||(x>=z&&y<=t)];

%t wknestcmpts[stn_]:=csm[Union[List/@stn,Select[Subsets[stn,{2}],wknXQ]]];

%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}]],Union@@#==Range[n]&&Length[wknestcmpts[#]]<=1&]],{n,0,5}]

%Y The binomial transform is the non-covering case A326338.

%Y The non-weak case is A326331.

%Y Simple graphs whose nesting edges are connected are A326330.

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

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

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Jun 28 2019