A250001 -id:A250001 - cp's OEIS frontend

A090338 Number of ways of arranging n straight lines in general position in the (affine) plane.

1, 1, 1, 1, 1, 6, 43, 922, 38609, 3111341

Offset: 0

Jon Wild and Laurence Reeves, Jan 27 2004

This is in the affine plane, rather than the projective plane, so two lines are either parallel or meet in one point.
Here we only consider arrangements of n lines in "general position", with every two lines meeting in one point and every intersection point lying on exactly two lines. See A241600 for the general case.
Two arrangements are considered the same if the lines in each arrangement can be numbered from 1 to n in such a way that, on each line, the order of crossings with the other lines is the same in the two arrangements. In particular, turning over the whole arrangement is allowed. (This does not imply that one arrangement can be continuously changed to the other (possibly after turning over) while keeping all lines straight, without changing the multiplicity of intersection points, and without a line passing through an intersection point, see the papers by Suvorov, Jaggi et al., Richter-Gebert, and Tsukamoto.)
Old name was "Number of full n-flups". The full n-flups are the topologically distinct planar configurations of n straight lines such that each line crosses each other line at exactly one intersection point and no two intersection points coincide.
Also, the number of distinct ways to divide a pancake with n straight cuts that result in the maximal number of pieces (see A000124, A000125).

			See illustration of a(5), the full pentaflups. Of the six, the last shown does not have reflectional symmetry, but we do not count its mirror image as distinct. All six are drawn with lines at equally-spaced angles; it is usually (but not always) possible to achieve this (41 out of 43 of the full 6-flups, for example, have equi-angled drawings)

Tobias Christ, Database of Combinatorially Different Simple Line Arrangements
Beat Jaggi, Peter Mani-Levitska, Bernd Sturmfels, and Neil White, Uniform oriented matroids without the isotopy property. Discrete Comput Geom 4, 97-100 (1989).
Jürgen Richter-Gebert, Two interesting oriented matroids, Documenta Mathematica 1 (1996), 137-148.
P. Suvorov, Isotopic but not rigidly isotopic plane systems of straight lines. In: Viro, O.Y., Vershik, A.M. (eds.) Topology and Geometry — Rohlin Seminar. Lecture Notes in Mathematics, vol 1346. Springer, Berlin, Heidelberg, pp. 545-556 (1988).
Yasuyuki Tsukamoto, New examples of oriented matroids with disconnected realization spaces (2012)
Jon Wild and Laurence Reeves, Illustration for a(5) = 6.

Cf. A000124, A000125, A090339 (when the lines need not be straight), A241600, A250001.

Edited by Max Alekseyev, May 15 2014
Further edits by N. J. A. Sloane, May 16 2014
a(9) from Christ added, and comments corrected by Günter Rote, Apr 14 2025

A241600 Number of ways of arranging n lines in the (affine) plane.

Original entry on oeis.org

1, 1, 2, 4, 9, 47, 791, 37830

Offset: 0

Max Alekseyev and N. J. A. Sloane, May 15 2014

This is in the affine plane, rather than the projective plane, so lines are either parallel or meet in one point.
Two arrangements are considered the same if one can be continuously changed to the other while keeping all lines straight, without changing the multiplicity of intersection points, and without a line passing through an intersection point. Turning over is also allowed.
a(n) might be called the size of the moduli space of n lines in the affine plane.
The subsequence giving the number of arrangements G_n of n lines in "general position" (with every two lines meeting in one point and every intersection point lying on exactly two lines) is given by A090338.
The moduli space of n points in the affine plane has been studied by several people (see for example Haiman and Miller, 2004; Martin, 2003). There is no direct connection with this problem, but these references are included for background information. - N. J. A. Sloane, Sep 13 2014
Lukas Finschi points out (email, Sep 19 2014) that a(n) = A063859(n)+1 for n <= 7 (but not for larger n). - N. J. A. Sloane, Sep 20 2014

			Let P_n = n parallel lines, S_n = star of n lines through a point, G_n = n lines in general position, L = P_1 = S_1 = G_1 = a single line.
a(1) = 1: L.
a(2) = 2: P_2, S_2.
a(3) = 4: P_3, P_2 L, S_3, G_3.
See link for illustrations of first 5 terms.

B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 4.

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.
Stefan Forcey, Planes and axioms, Univ. Akron (2024). See p. 3.
Stefan Forcey, Counting plane arrangements via oriented matroids, arXiv:2504.11461 [math.HO], 2025. See pp. 5, 18.
Komei Fukuda, Hiroyuki Miyata, and Sonoko Moriyama Complete Enumeration of Small Realizable Oriented Matroids, arXiv:1204.0645 [math.CO], 2012; Discrete Comput. Geom. 49 (2013), no. 2, 359--381. MR3017917. (Further background information.)
Mark Haiman, with an Appendix by Ezra Miller, Commutative algebra of n points in the plane. Trends Commut. Algebra, MSRI Publ 51 (2004): 153-180. (Background)
Sergey Kalmykov, Isolated visible infinite straight lines and their combinations, 1920-1922, private collection, on display at Tretyakov gallery. [illustrates a(1)-a(4), and part of a(5)]
J. L. Martin, The slopes determined by n points in the plane. (Background)
Jeremy L. Martin, The slopes determined by n points in the plane, arXiv:math/0302106 [math.AG], 2003-2006; Duke Math. J. 131 (2006), no. 1, 119-165. (Background)
N. J. A. Sloane, Illustration of a(1)-a(5)
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (2015 talk slides)

Cf. A090338 (lines in general position), A090339 (curved lines in general position), A250001 (circles).

Cf. also A063859, A003036, A048872, A048873, A132346.

a(n) >= A000041(n). - Pablo Hueso Merino, May 10 2021

a(6) and a(7) from Lukas Finschi, Sep 19 2014

A249752 Triangle read by rows: T(n,k) = number of arrangements of n circles in the affine plane having k outer circles.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 14, 52, 60, 47

Offset: 1

Omar E. Pol, Dec 14 2014

Consider the arrangements of n circles described in A250001.
T(n,k) is the number of arrangements of n circles in the affine plane whose perimeters are formed with parts from k circles. - Omar E. Pol, Aug 09 2015
From Omar E. Pol, Jul 06 2016: (Start)
Observation 1: column 1 gives A250001, at least if 1<=n<=4.
Observation 2: sum of n-th row = T(n+1,1), at least if 1<=n<=3. (End)

			Triangle begins:
1;
1,   2;
3,   5,  6;
14, 52, 60, 47;
...

Sum of n-th row = A250001(n).
Leading diagonal gives A274702.
Cf. A252158, A261070, A274776, A274777.

Corrected a(5)-a(6) and added a(7)-a(10) from Omar E. Pol, Oct 13 2015

A252158 Triangle read by rows, 1 <= k <= n, T(n,k) = number of arrangements of n circles in the affine plane having k solid regions in which the union of solid circles is connected.

Original entry on oeis.org

1, 2, 1, 11, 2, 1, 155, 15, 2, 1

Offset: 1

Omar E. Pol, Dec 14 2014

Consider the rules for the arrangements of n circles described in A250001.
Right border gives A000012. - Omar E. Pol, Aug 09 2015
From Omar E. Pol, May 21 2017: (Start)
Note that T(4,1) = 155 does not include the arrangement of four circles in which there is a central circle that is surrounded by the union of three circles, because in that arrangement there are two solid regions, not one. The smallest solid region is formed by the central solid circle. The largest solid region is formed by the union of three solid circles.
T(4,2) = 15 includes the arrangement mentioned above.
1 together with the first column gives A287149.
For another version see A285996. (End) (Comment clarified on Jun 15 2017.)

			Triangle begins:
1;
2,   1;
11,  2,   1;
155, 15,  2,  1;

Row sums give A250001, n >= 1.

Cf. A249752, A285996 (another version), A287149.

Clarified definition and a(7)-a(10) added by Omar E. Pol, May 21 2017

Clarified definition and comment by Omar E. Pol, Jun 15 2017

A383083 The number of distinct straightedge-and-compass constructions that can be made with no lines and n circles.

Original entry on oeis.org

1, 2, 1, 4, 44, 1084, 91192

Offset: 0

Peter Kagey, Apr 16 2025

A straightedge-and-compass construction starts with 2 points marked on the plane, traditionally (0,0) and (1,0). In the constructions counted by this sequence, only the compass is used. Circles can be drawn at any marked point through any other marked point, and new points are marked where circles intersect.

Wikipedia, Mohr-Mascheroni theorem
Peter Kagey, Illustration of a(0)=1, a(1)=2, a(2)=1, and a(3)=4.
Peter Kagey, Illustration of a(4)=44.

Cf. A250001, A333944, A339266, A383082.

A274776 Irregular triangle read by rows: T(n,k) = number of arrangements of n circles in the affine plane forming k regions, including the regions that do not belong to the circles.

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 4, 4, 2, 0, 4, 0, 0, 0

Offset: 1

Omar E. Pol, Jul 06 2016

Consider the arrangements of n circles described in A250001.

Note that the sum of the 4th row must be equal to A250001(4) = 173.

			Triangle begins:
1;
0, 2, 1;
0, 0, 4, 4, 2, 0, 4;
0, 0, 0, ...
...
For n = 3 and k = 5 there are 2 arrangements of 3 circles in the affine plane forming 5 regions, including the regions that do not belong to the circles, so T(3,5) = 2.
For n = 3 and k = 6 there are no arrangements of 3 circles in the affine plane forming 6 regions, including the regions that do not belong to the circles, so T(3,6) = 0.
Of course, there is a right triangle of all zeros starting from the second row.

Sum of n-th row = A250001(n).
First differs from A274777 at a(10).
Cf. A250001, A249752, A252158, A261070, A274818, A274822.

T(n,k) = A274818(n,k)/k.

A274777 Irregular triangle read by rows: T(n,k) = number of arrangements of n circles in the affine plane forming k regions, excluding the regions that do not belong to the circles.

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 4, 4, 2, 1, 3, 0, 0, 0

Offset: 1

Omar E. Pol, Jul 06 2016

In other words: not counting the regions between circles.
Consider the arrangements of n circles described in A250001.
Note that the sum of the 4th row must be equal to A250001(4) = 173.

			Triangle begins:
1;
0, 2, 1;
0, 0, 4, 4, 2, 1, 3;
0, 0, 0, ...
...
For n = 3 and k = 5 there are 2 arrangements of 3 circles in the affine plane forming 5 regions, excluding the regions that do not belong to the circles, so T(3,5) = 2.
For n = 3 and k = 6 there is only one arrangement of 3 circles in the affine plane forming 6 regions, excluding the regions that do not belong to the circles, so T(3,6) = 1.
Of course, there is a right triangle of all zeros starting from the second row.

Sum of n-th row = A250001(n).
First differs from A274776 at a(10).
Cf. A250001, A249752, A252158, A261070, A274819, A274823.

T(n,k) = A274819(n,k)/k.

A288559 Number of arrangements of n pseudo-circles in the affine plane.

Original entry on oeis.org

1, 1, 3, 14, 173, 16977, 17552169

Offset: 0

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

These counts have been reduced for mirror symmetry.
See A250001, the main entry for this problem, for further information.
This sequence is also an upper bound for A250001. - Andrii Shportko, Jun 03 2025

Cf. A250001, A275923, A275924, A288554-A288568.

After consulting with Jon Wild, a(6) added by Andrii Shportko, Jun 03 2025

A274818 Triangle read by rows: T(n,k) = total number of regions in all arrangements of n circles in the affine plane forming k regions, including the regions that do not belong to the circles.

Original entry on oeis.org

1, 0, 4, 3, 0, 0, 12, 16, 10, 0, 28, 0, 0, 0

Offset: 1

Omar E. Pol, Jul 07 2016

Consider the arrangements of n circles described in A250001.

			Triangle begins:
1;
0, 4,  3;
0, 0, 12, 16, 10, 0, 28;
0, 0,  0, ...
...
For n = 3 and k = 5 there are 2 arrangements of 3 circles in the affine plane forming 5 regions, including the regions that do not belong to the circles, so T(3,5) = 2*5 = 10.
For n = 3 and k = 6 there are no arrangements of 3 circles in the affine plane forming 6 regions, including the regions that do not belong to the circles, so T(3,6) = 0*5 = 0.
Of course, there is a right triangle of all zeros starting from the second row.

Row sums give A274822.
First differs from A274819 at a(10).
Cf. A250001, A274776.

T(n,k) = k*A274776(n,k).

A274819 Triangle read by rows: T(n,k) = total number of regions in all arrangements of n circles in the affine plane forming k regions, excluding the regions that do not belong to the circles.

Original entry on oeis.org

1, 0, 4, 3, 0, 0, 12, 16, 10, 6, 21, 0, 0, 0

Offset: 1

Omar E. Pol, Jul 07 2016

In other words: not counting the regions between circles.

Consider the arrangements of n circles described in A250001.

			Triangle begins:
1;
0, 4,  3;
0, 0, 12, 16, 10, 6, 21;
0, 0,  0, ...
...
For n = 3 and k = 5 there are 2 arrangements of 3 circles in the affine plane forming 5 regions, excluding the regions that do not belong to the circles, so T(3,5) = 2*5 = 10.
For n = 3 and k = 6 there is only one arrangement of 3 circles in the affine plane forming 6 regions, excluding the regions that do not belong to the circles, so T(3,6) = 1*6 = 6.
Of course, there is a right triangle of all zeros starting from the second row.

Row sums give A274823(n).
First differs from A274818 at a(10).
Cf. A250001, A274777.

T(n,k) = k*A274777(n,k).

cp's OEIS Frontend

A090338 Number of ways of arranging n straight lines in general position in the (affine) plane.

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A241600 Number of ways of arranging n lines in the (affine) plane.

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A249752 Triangle read by rows: T(n,k) = number of arrangements of n circles in the affine plane having k outer circles.

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A252158 Triangle read by rows, 1 <= k <= n, T(n,k) = number of arrangements of n circles in the affine plane having k solid regions in which the union of solid circles is connected.

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A383083 The number of distinct straightedge-and-compass constructions that can be made with no lines and n circles.

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A274776 Irregular triangle read by rows: T(n,k) = number of arrangements of n circles in the affine plane forming k regions, including the regions that do not belong to the circles.

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A274777 Irregular triangle read by rows: T(n,k) = number of arrangements of n circles in the affine plane forming k regions, excluding the regions that do not belong to the circles.

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A288559 Number of arrangements of n pseudo-circles in the affine plane.

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A274818 Triangle read by rows: T(n,k) = total number of regions in all arrangements of n circles in the affine plane forming k regions, including the regions that do not belong to the circles.

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A274819 Triangle read by rows: T(n,k) = total number of regions in all arrangements of n circles in the affine plane forming k regions, excluding the regions that do not belong to the circles.

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