A326244 Number of labeled n-vertex simple graphs without crossing or nesting edges.

1, 1, 2, 8, 36, 160, 704, 3088, 13536, 59328, 260032, 1139712

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..11.

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.

Gus Wiseman, The a(4) = 36 non-crossing, non-nesting simple labeled graphs.

Formula

A006125(n) = a(n) + A326279(n).

Conjectures from Colin Barker, Jun 28 2019: (Start)

G.f.: (1 - x)*(1 - 4*x) / (1 - 6*x + 8*x^2 - 4*x^3).

a(n) = 6*a(n-1) - 8*a(n-2) + 4*a(n-3) for n>2.

(End)

Mathematica

croXQ[stn_]:=MatchQ[stn, {___, {x_, y_}, ___, {z_, t_}, ___}/; x<z<y<t||z<x<t<y];
nestQ[stn_]:=MatchQ[stn, {___, {x_, y_}, ___, {z_, t_}, ___}/; x<z<t<y||z<x<y<t];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], !croXQ[#]&&!nestQ[#]&]], {n, 0, 5}]

Crossrefs

The case for set partitions is A001519.

Simple graphs with crossing or nesting edges are A326279.

Cf. A000088, A000108, A006125, A016098, A054726, A095661, A117662.

Cf. A326209, A326210, A326250, A326255, A326256.

Sequence in context: A228791 A088675 A228197 * A027743 A152124 A147722

Adjacent sequences: A326241 A326242 A326243 * A326245 A326246 A326247

Keyword

nonn,more

Author

Gus Wiseman, Jun 20 2019

Status

approved