A000241 -id:A000241 - cp's OEIS frontend

A276110 The number of rotation systems of drawings of the complete graph K_n, where the rotation system describes the clockwise cyclic order of incident edges around each vertex.

1, 2, 5, 102, 11556, 5370725, 7198391729

Offset: 3

Manfred Scheucher, Aug 18 2016

The number of realizable order types on n points in the plane (A063666) is exactly the number of rotation systems of straight-line drawings of K_n.

B. M. Ábrego, O. Aichholzer, S. Fernández-Merchant, T. Hackl, J. Pammer, A. Pilz, P. Ramos, G. Salazar, and B. Vogtenhuber, All Good Drawings of Small Complete Graphs, In Proc. 31st European Workshop on Computational Geometry EuroCG '15, pages 57-60, Ljubljana, Slovenia, 2015.
A. Arroyo, D. McQuillan, and B. Richter, Drawings of Kn with the same rotation scheme are the same up to Reidemeister moves (Gioan's Theorem), submitted, 2015.
J. Kynčl, Enumeration of simple complete topological graphs, European Journal of Combinatorics, 30(7):1676-1685, 2009.
Wikipedia, Rotation Systems

Coincides with A276109 for n <= 5.

Cf. A000241, A063666.

A226917 Number of non-sphere-homeomorphic crossing optimal 2-page book drawings of complete graph K_{2n+1}.

Original entry on oeis.org

1, 4, 9, 25, 58, 142, 324, 748, 1672, 3736, 8208, 17968, 38944, 84064, 180288, 385216, 819328

Offset: 2

N. J. A. Sloane, Jul 01 2013

Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, Gelasio Salazar, The 2-Page Crossing Number of K_n, arXiv:1206.5669 [math.CO], 2012.
Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, Gelasio Salazar, The 2-Page Crossing Number of K_n, Discrete Comput. Geom. 49 (2013), no. 4, 747--777. MR3068573

Cf. A000241.

Conjecture: a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 8*a(n-4); g.f.: -x^2*(5*x^3-5*x^2+1) / ((2*x-1)^2*(2*x^2-1)). - Colin Barker, May 08 2014

A250221 Least k such that A_n(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 11, -1, 16, 12, -1, 9, -1, 11, -1, -1, -1, 11, -1, 8, -1, -1, 126, -1, -1, -1, -1, -1, -1, 26, 27, -1, -1, -1, -1, -1, -1, -1, -1, 29, 31, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 33, -1, -1, -1, -1, -1, -1, 19, -1, -1, 45, -1, -1, -1, -1, -1, -1, 35, -1, -1, 8

Offset: 1

Eric Chen, Dec 24 2014

sign

a(A053169(n)) = -1, but what is a(53169)?
a(319) is the first unknown term. (See A000319)
a(241) should be -1. (See A000241)

Cf. A053873, A053169.

cp's OEIS Frontend

A276110 The number of rotation systems of drawings of the complete graph K_n, where the rotation system describes the clockwise cyclic order of incident edges around each vertex.

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A226917 Number of non-sphere-homeomorphic crossing optimal 2-page book drawings of complete graph K_{2n+1}.

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A250221 Least k such that A_n(k) = n, or -1 if no such k exists.

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