A006248 -id:A006248 - cp's OEIS frontend

A288568 Number of non-isomorphic connected arrangements of n pseudo-circles on a sphere, in the sense that the union of the pseudo-circles is a connected set, reduced for mirror symmetry.

1, 1, 1, 3, 21, 984, 609423

Offset: 0

N. J. A. Sloane, Jun 13 2017, based on information supplied by Jon Wild on August 31 2016

These counts have been reduced for mirror symmetry. Computed up to n=5 by Jon Wild and Christopher Jones and communicated to N. J. A. Sloane on August 31 2016. Definition corrected Dec 10 2017 thanks to Manfred Scheucher, who has computed same result with Stefan Felsner independently.
The list of arrangements is available online on the Homepage of Pseudocircles (see below) and a detailed description for the enumeration can be found in Arrangements of Pseudocircles: On Circularizability (see below). - Manfred Scheucher, Dec 11 2017
See A250001, the main entry for this problem, for further information.

S. Felsner and M. Scheucher Homepage of Pseudocircles
S. Felsner and M. Scheucher, Arrangements of Pseudocircles: On Circularizability, arXiv:1712.02149 [cs.CG], 2017.

Cf. A250001, A275923, A275924, A288554-A288568, A296406, A296407-A296412, A006248.

a(n) = 2^(\Theta(n^2)). (cf. Arrangements of Pseudocircles: On Circularizability)

a(6) from Manfred Scheucher, Dec 11 2017

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.

Original entry on oeis.org

1, 1, 1, 2, 3, 16, 135, 3315, 158830, 14320182, 2343203071, 691470685682, 366477801792538

Offset: 1

N. J. A. Sloane

Also the number of nonisomorphic nondegenerate acyclic rank 3 oriented matroids on n elements. - Manfred Scheucher, May 09 2022

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

Cf. A006248, A014540, A063852, A063666, A325595, A325628, A006245

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

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

A296407 Number of digon-free connected arrangements of n pseudo-circles on a sphere, in the sense that the union of the pseudo-circles is a connected set, reduced for mirror symmetry.

Original entry on oeis.org

1, 1, 1, 1, 3, 30, 4509

Offset: 0

Manfred Scheucher, Dec 11 2017

For more information, see A288568.

Cf. A250001, A275923, A275924, A288554-A288568, A296406, A296407-A296412, A006248.

A296406 Number of non-isomorphic arrangements of n pairwise intersecting pseudo-circles on a sphere, reduced for mirror symmetry.

Original entry on oeis.org

1, 1, 1, 2, 8, 278, 145058, 447905202

Offset: 0

Manfred Scheucher, Dec 11 2017

The list of arrangements is available online on the Homepage of Pseudocircles (see below) and a detailed description for the enumeration can be found in Arrangements of Pseudocircles: On Circularizability (see below).

S. Felsner and M. Scheucher Homepage of Pseudocircles
S. Felsner and M. Scheucher, Arrangements of Pseudocircles: On Circularizability, arXiv:1712.02149 [cs.CG], 2017.
Yan Alves Radtke, Stefan Felsner, Johannes Obenaus, Sandro Roch, Manfred Scheucher, and Birgit Vogtenhuber, Flip Graph Connectivity for Arrangements of Pseudolines and Pseudocircles, arXiv:2310.19711 [math.CO], 2023. See p. 41.

Cf. A250001, A275923, A275924, A288554-A288568, A296407-A296412, A006248.

a(n) = 2^(\Theta(n^2)). (cf. Arrangements of Pseudocircles: On Circularizability)

A296412 Number of non-isomorphic digon-free cylindrical arrangements of n pairwise intersecting pseudo-circles on a sphere, in the sense that two cells of the arrangement are separated by each of the pseudo-circles, reduced for mirror symmetry.

Original entry on oeis.org

1, 1, 1, 1, 2, 14, 2131, 3012906

Offset: 0

Manfred Scheucher, Dec 11 2017

For more information, see A296406.

Cf. A250001, A275923, A275924, A288554-A288568, A296406, A296407-A296412, A006248.

A018242 Number of projective order types.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 11, 135, 4381, 312114, 41693377

Offset: 0

N. J. A. Sloane

Table 5.6.1 in the Felsner-Goodman survey contains this sequence in the second row, but the line is incorrectly labeled. The origin of these data is the paper of Aichholzer and Krasser. - Günter Rote, Apr 16 2025

Oswin Aichholzer, Hannes Krasser: Abstract order type extension and new results on the rectilinear crossing number. Comput. Geom. 36(1), 2-15 (2007), Table 1
Stefan Felsner and Jacob E. Goodman, Pseudoline Arrangements, Chapter 5 of Handbook of Discrete and Computational Geometry, 3rd edition, Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, CRC Press, 2017, see Table 5.6.1. [alternative link][alternative link] [Specific reference for this sequence] - N. J. A. Sloane, Nov 14 2023
Komei Fukuda, Hiroyuki Miyata, Sonoko Moriyama, Complete Enumeration of Small Realizable Oriented Matroids. Discrete Comput. Geom. 49 (2013), no. 2, 359-381, see Table 2. MR3017917. Also arXiv:1204.0645 [math.CO], 2012. - From _N. J. A. Sloane_, Feb 16 2013

Cf. A006247, A006248, A063666. A diagonal of A222317.

Asymptotics: a(n) = 2^(Theta(n log n)). 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 log n) <= a(n) <= 2^(d n log n) is satisfied. For more information see e.g. the Handbook of Discrete and Computational Geometry. - Manfred Scheucher, Sep 12 2019

a(11) from Franz Aurenhammer (auren(AT)igi.tu-graz.ac.at), Feb 05 2002

A063851 Triangle T(n,k) (n >= 3, k = 1..n-2) read by rows giving number of nonisomorphic nondegenerate oriented matroids with n points in n-k dimensions.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 11, 11, 1, 1, 1, 135, 2628, 135, 1, 1, 1, 4382, 9276601, 9276601, 4382, 1, 1, 1, 312356

Offset: 3

N. J. A. Sloane, Aug 26 2001

			Triangle begins:
1
1,1,
1,1,1,
1,1,1,4,
1,1,1,11,11,
1,1,1,135,2628,135,
1,1,1,4382,9276601,9276601,4382,
1,1,1,312356,...

Lukas Finschi, Homepage of Oriented Matroids [Gives T(9, 5) = T(9, 6) = 9276595.]
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.
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.
Komei Fukuda, Hiroyuki Miyata and Sonoko Moriyama, Complete Enumeration of Small Realizable Oriented Matroids. Discrete Comput. Geom. 49 (2013), no. 2, 359--381. MR3017917. - From _N. J. A. Sloane_, Feb 16 2013 [Beware typos in Table 1.]

For numbers when degenerate matroids are included see A063804. Two rightmost diagonals are A006248 and A222315. Row sums give A063852.

More terms taken from Fukuda et al., 2013. - N. J. A. Sloane, Feb 16 2013

A111072 Write the digit string 0123456789, repeated infinitely many times. Then, starting from the first "0" digit at the left end, move to the right by one digit (to the "1"), then two digits (to the "3"), then three digits (to the "6"), four digits ("0"), five digits ("5"), and so on. Partial sums of the digits thus reached are 0, 1, 4, 10, 10, 15, ...

Original entry on oeis.org

0, 1, 4, 10, 10, 15, 16, 24, 30, 35, 40, 46, 54, 55, 60, 60, 66, 69, 70, 70, 70, 71, 74, 80, 80, 85, 86, 94, 100, 105, 110, 116, 124, 125, 130, 130, 136, 139, 140, 140, 140, 141, 144, 150, 150, 155, 156, 164, 170, 175, 180, 186, 194, 195, 200, 200, 206, 209, 210

Offset: 0

Giorgio Balzarotti and Paolo P. Lava, Oct 10 2005

The first differences 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, etc. are in A008954.

			a(9) = 35 because a(8) - a(7) + (9 mod 10) = 30 - 24 + 9 = 15 and a(8) + (15 mod 10) = 30 + 5 = 35.
Jumping we move to the numbers 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, 5, 1, 8, 6, etc. Summing the numbers we obtain 0, 0+1 = 1, 1+3 = 4, 4+6 = 10, 10+0 = 10, 10+5 = 16, etc.

Giorgio Balzarotti and Paolo P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 62.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
J. Bokowski & N. J. A. Sloane, Emails, June 1994.

Cf. A008954.

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
      [0,1,3,6,0,5,1,8,6,5,5,6,8,1,5,0,6,3,1,0,0]
      [1+irem(n, 20)])
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 23 2021

Fold[Append[#1, #1[[-1]] + Mod[(#1[[-1]] - #1[[-2]] + Mod[#2, 10]), 10]] &, {0, 1}, Range[2, 58]] (* Michael De Vlieger, Nov 05 2017 *)

a(n+1) = a(n) + (a(n) - a(n-1) + (n+1) mod 10) mod 10, with a(0)=0, a(1)=1.

G.f.: x*(x^12+3*x^11+6*x^10+5*x^8+5*x^6+5*x^4+6*x^2+3*x+1) / (x^16 -x^15 -x^11 +x^10 +x^6 -x^5 -x +1). - Alois P. Heinz, Jan 23 2021

A180500 Triangle read by row. T(n,m) gives the number of isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the projective plane.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 13, 1, 5, 48, 626, 6570, 1, 25, 1329, 86715, 4822394, 181403533

Offset: 0

Vincent Pilaud, Sep 08 2010

J. Ferté, V. Pilaud and M. Pocchiola, On the number of arrangements of five double pseudolines, Abstracts 18th Fall Workshop on Comput. Geom. (FWCG08), Troy, NY, October 2008.

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.

See A180501 for isomorphism classes of all (not only simple) arrangements of n pseudolines and m double pseudolines in the projective plane.
See A180503 for isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the Moebius strip.
First diagonal gives A006248.

A296408 Number of cylindrical connected arrangements of n pseudo-circles on a sphere, in the sense that the union of the pseudo-circles is a connected set and two cells of the arrangement are separated by each of the pseudo-circles, reduced for mirror symmetry.

Original entry on oeis.org

1, 1, 1, 3, 20, 900, 530530

Offset: 0

Manfred Scheucher, Dec 11 2017

For more information, see A288568.

Cf. A250001, A275923, A275924, A288554-A288568, A296406, A296407-A296412, A006248.

cp's OEIS Frontend

A288568 Number of non-isomorphic connected arrangements of n pseudo-circles on a sphere, in the sense that the union of the pseudo-circles is a connected set, reduced for mirror symmetry.

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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.

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A296407 Number of digon-free connected arrangements of n pseudo-circles on a sphere, in the sense that the union of the pseudo-circles is a connected set, reduced for mirror symmetry.

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A296406 Number of non-isomorphic arrangements of n pairwise intersecting pseudo-circles on a sphere, reduced for mirror symmetry.

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A296412 Number of non-isomorphic digon-free cylindrical arrangements of n pairwise intersecting pseudo-circles on a sphere, in the sense that two cells of the arrangement are separated by each of the pseudo-circles, reduced for mirror symmetry.

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A018242 Number of projective order types.

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A063851 Triangle T(n,k) (n >= 3, k = 1..n-2) read by rows giving number of nonisomorphic nondegenerate oriented matroids with n points in n-k dimensions.

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A180500 Triangle read by row. T(n,m) gives the number of isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the projective plane.

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A296408 Number of cylindrical connected arrangements of n pseudo-circles on a sphere, in the sense that the union of the pseudo-circles is a connected set and two cells of the arrangement are separated by each of the pseudo-circles, reduced for mirror symmetry.

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