A326279 Number of labeled n-vertex simple graphs containing either a crossing or a nesting pair of edges.

0, 0, 0, 0, 28, 864, 32064, 2094064

Offset

0,5

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..7.

Formula

A006125(n) = a(n) + A326244(n).

Example

The a(4) = 28 edge-sets:
  {13,24}  {12,13,24}  {12,13,14,23}  {12,13,14,23,24}  {12,13,14,23,24,34}
  {14,23}  {12,14,23}  {12,13,14,24}  {12,13,14,23,34}
           {13,14,23}  {12,13,23,24}  {12,13,14,24,34}
           {13,14,24}  {12,13,24,34}  {12,13,23,24,34}
           {13,23,24}  {12,14,23,24}  {12,14,23,24,34}
           {13,24,34}  {12,14,23,34}  {13,14,23,24,34}
           {14,23,24}  {13,14,23,24}
           {14,23,34}  {13,14,23,34}
                       {13,14,24,34}
                       {13,23,24,34}
                       {14,23,24,34}

Mathematica

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

Crossrefs

Crossing and nesting simple graphs are (both) A326210, while non-crossing, non-nesting simple graphs are A326244.

Cf. A000108, A001519, A006125, A016098, A054726, A095661, A324170.

Cf. A326209, A326211, A326248, A326250, A326256.

Sequence in context: A214133 A119055 A324450 * A278195 A095656 A057412

Adjacent sequences: A326276 A326277 A326278 * A326280 A326281 A326282

Keyword

nonn,more

Author

Gus Wiseman, Jun 23 2019

Status

approved