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

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

Offset

0,4

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

Table of n, a(n) for n=0..6.

Gus Wiseman, The a(4) = 29 covering simple graphs whose weakly nesting edges are connected.

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}]

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.

Sequence in context: A210827 A092251 A304553 * A331389 A243435 A064570

Adjacent sequences: A326334 A326335 A326336 * A326338 A326339 A326340

Keyword

nonn,more

Author

Gus Wiseman, Jun 28 2019

Status

approved