A326248 -id:A326248 - cp's OEIS frontend

A326289 a(0) = 0, a(n) = 2^binomial(n,2) - 2^(n - 1).

0, 0, 0, 4, 56, 1008, 32736, 2097088, 268435328, 68719476480, 35184372088320, 36028797018962944, 73786976294838204416, 302231454903657293672448, 2475880078570760549798240256, 40564819207303340847894502555648, 1329227995784915872903807060280311808

Offset: 0

Gus Wiseman, Jun 23 2019

nonn

Number of simple graphs with vertices {1..n} containing two edges {a,b}, {c,d} that are weakly crossing, meaning a <= c < b <= d or c <= a < d <= b.

			The a(4) = 56 weakly crossing edge-sets:
  {12,13}  {12,13,14}  {12,13,14,23}  {12,13,14,23,24}  {12,13,14,23,24,34}
  {12,14}  {12,13,23}  {12,13,14,24}  {12,13,14,23,34}
  {12,23}  {12,13,24}  {12,13,14,34}  {12,13,14,24,34}
  {12,24}  {12,13,34}  {12,13,23,24}  {12,13,23,24,34}
  {12,34}  {12,14,23}  {12,13,23,34}  {12,14,23,24,34}
  {13,14}  {12,14,24}  {12,13,24,34}  {13,14,23,24,34}
  {13,23}  {12,14,34}  {12,14,23,24}
  {13,24}  {12,23,24}  {12,14,23,34}
  {13,34}  {12,23,34}  {12,14,24,34}
  {14,24}  {12,24,34}  {12,23,24,34}
  {14,34}  {13,14,23}  {13,14,23,24}
  {23,24}  {13,14,24}  {13,14,23,34}
  {23,34}  {13,14,34}  {13,14,24,34}
  {24,34}  {13,23,24}  {13,23,24,34}
           {13,23,34}  {14,23,24,34}
           {13,24,34}
           {14,23,24}
           {14,23,34}
           {14,24,34}
           {23,24,34}

Cf. A000088, A000108, A002662, A006125, A016098, A054726, A324170.

Cf. A326210, A326244, A326248, A326250, A326257, A326279, A326290.

Table[If[n==0,0,2^Binomial[n,2]-2^(n-1)],{n,0,5}]

A326278 Number of n-vertex, 2-edge multigraphs that are not nesting. Number of n-vertex, 2-edge multigraphs that are not crossing.

Original entry on oeis.org

0, 0, 1, 9, 34, 90, 195, 371, 644, 1044, 1605, 2365, 3366, 4654, 6279, 8295, 10760, 13736, 17289, 21489, 26410, 32130, 38731, 46299, 54924, 64700, 75725, 88101, 101934, 117334, 134415, 153295, 174096, 196944, 221969, 249305, 279090, 311466, 346579, 384579

Offset: 0

Gus Wiseman, Jun 23 2019

nonn

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.

			The a(3) = 9 non-crossing multigraphs:
  {12,12}
  {12,13}
  {12,23}
  {13,12}
  {13,13}
  {13,23}
  {23,12}
  {23,13}
  {23,23}

A326247(n) <= a(n) <= A000537(n).
The case for 2-edge simple graphs (rather than multigraphs) is A117662.
Cf. A000108, A001519, A006125, A016098, A054726, A095661.
Cf. A326210, A326243, A326244, A326248, A326250.

croXQ[stn_]:=MatchQ[stn,{_,{x_,y_},_,{z_,t_},_}/;x

Conjectures from Colin Barker, Jun 25 2019: (Start)
G.f.: x^2*(1 + 4*x - x^2) / (1 - x)^5.
a(n) = (n*(3 - 4*n + n^3)) / 6 .
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
(End)

cp's OEIS Frontend

A326289 a(0) = 0, a(n) = 2^binomial(n,2) - 2^(n - 1).

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