A006247 Number of simple arrangements of n pseudolines in the projective plane with a marked cell. Number of Euclidean pseudo-order types: nondegenerate abstract order types of configurations of n points in the plane.
1, 1, 1, 2, 3, 16, 135, 3315, 158830, 14320182, 2343203071, 691470685682, 366477801792538
Offset: 1
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- O. Aichholzer, Order Types for Small Point Sets
- O. Aichholzer, F. Aurenhammer and H. Krasser, Enumerating order types for small point sets with applications, In Proc. 17th Ann. ACM Symp. Computational Geometry, pages 11-18, Medford, Massachusetts, USA, 2001. [Computes a(10)]
- Stefan Felsner and Jacob E. Goodman, Pseudoline Arrangements, Chapter 5 of Handbook of Discrete and Computational Geometry, CRC Press, 2017, see Table 5.6.1. [Specific reference for this sequence] - _N. J. A. Sloane_, Nov 14 2023
- J. Ferté, V. Pilaud and M. Pocchiola, On the number of simple arrangements of five double pseudolines, arXiv:1009.1575 [cs.CG], 2010; Discrete Comput. Geom. 45 (2011), 279-302.
- Lukas Finschi, A Graph Theoretical Approach for Reconstruction and Generation of Oriented Matroids, A dissertation submitted to the Swiss Federal Institute of Technology, Zurich for the degree of Doctor of Mathematics, 2001.
- Lukas Finschi, Homepage of Oriented Matroids
- L. Finschi and K. Fukuda, Complete combinatorial generation of small point set configurations and hyperplane arrangements, pp. 97-100 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
- Henry Förster, Philipp Kindermann, Tillmann Miltzow, Irene Parada, Soeren Terziadis, and Birgit Vogtenhuber, Geometric Thickness of Multigraphs is (exists in reals)-complete, arXiv:2312.05010 [cs.CG], 2023.
- Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry [alternative link], CRC Press, 2017, see Table 5.6.1. [General reference for 2017 edition of the Handbook] - _N. J. A. Sloane_, Nov 14 2023
- J. E. Goodman and R. Pollack, Semispaces of configurations, cell complexes of arrangements, J. Combin. Theory, A 37 (1984), 257-293.
- D. E. Knuth, Axioms and hulls, Lect. Notes Comp. Sci., Vol. 606.
- Alexander Pilz and Emo Welzl, Order on order types, Discrete & Computational Geometry, 59 (No. 4, 2015), 886-922.
- Manfred Scheucher, Hendrik Schrezenmaier, and Raphael Steiner, A Note On Universal Point Sets for Planar Graphs, arXiv:1811.06482 [math.CO], 2018.
Formula
Asymptotics: a(n) = 2^(Theta(n^2)). This is Bachmann-Landau notation, that is, there are constants n_0, c, and d, such that for every n >= n_0 the inequality 2^{c n^2} <= a(n) <= 2^{d n^2} is satisfied. For more information see e.g. the Handbook of Discrete and Computational Geometry. - Manfred Scheucher, Sep 12 2019
Extensions
a(11) from Franz Aurenhammer (auren(AT)igi.tu-graz.ac.at), Feb 05 2002
a(12) from Manfred Scheucher and Günter Rote, Aug 31 2019
a(13) from Manfred Scheucher and Günter Rote, Sep 12 2019
Definition clarified by Manfred Scheucher, Jun 22 2023
Comments