A326210 Number of labeled simple graphs with vertices {1..n} containing a nesting pair of edges, where two edges {a,b}, {c,d} are nesting if a < c and b > d or a > c and b < d.

0, 0, 0, 0, 16, 672, 29888, 2071936, 268204288, 68717285888, 35184350796800, 36028796807919616, 73786976292712960000, 302231454903635611721728, 2475880078570760326175178752, 40564819207303340845566684397568, 1329227995784915872903782635437883392

Offset: 0

Author

Gus Wiseman, Jun 19 2019

Keywords

• nonn

Comments

Also simple graphs containing a crossing pair of edges, where two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b.

Also simple graphs such that, if the edges are listed in lexicographic order, their maxima (seconds) are not weakly increasing.

Examples

			The a(4) = 16 nesting edge-sets:
  {14,23}
  {12,14,23}
  {13,14,23}
  {14,23,24}
  {14,23,34}
  {12,13,14,23}
  {12,14,23,24}
  {12,14,23,34}
  {13,14,23,24}
  {13,14,23,34}
  {14,23,24,34}
  {12,13,14,23,24}
  {12,13,14,23,34}
  {12,14,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}
The a(4) = 16 crossing edge-sets:
  {13,24}
  {12,13,24}
  {13,14,24}
  {13,23,24}
  {13,24,34}
  {12,13,14,24}
  {12,13,23,24}
  {12,13,24,34}
  {13,14,23,24}
  {13,14,24,34}
  {13,23,24,34}
  {12,13,14,23,24}
  {12,13,14,24,34}
  {12,13,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

Non-nesting graphs are A054726.

Nesting digraphs are A326209.

Nesting (or crossing) set partitions are A016098.

MM-numbers of nesting multiset partitions are A326256.

Cf. A001519, A006125, A095661, A324170, A324172, A324326.

Cf. A326211, A326243, A326244, A326248, A326250.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],!OrderedQ[Last/@#]&]],{n,0,5}]

PARI

seq(n)={my(p=1 + 3/2*x - x^2 - x/2*sqrt(1 - 12*x + 4*x^2 + O(x^n))); concat([0], vector(n, k, 2^binomial(k,2)-polcoef(p,k)))} \\ Andrew Howroyd, Aug 26 2019

Formula

A006125(n) = a(n) + A054726(n).

Extensions

Terms a(7) and beyond from Andrew Howroyd, Aug 26 2019