A006247 -id:A006247 - cp's OEIS frontend

A006245 Number of primitive sorting networks on n elements; also number of rhombic tilings of a 2n-gon.

1, 1, 2, 8, 62, 908, 24698, 1232944, 112018190, 18410581880, 5449192389984, 2894710651370536, 2752596959306389652, 4675651520558571537540, 14163808995580022218786390, 76413073725772593230461936736

Offset: 1

N. J. A. Sloane

Also the number of commutation classes of reduced words for the longest element of a Weyl group of type A_{n-1} (see Armstrong reference).
Also the number of signotopes of rank 3, i.e., mappings X:{{1..n} choose 3}->{+,-} such that for any four indices a < b < c < d, the sequence X(a,b,c), X(a,b,d), X(a,c,d), X(b,c,d) changes its sign at most once (see Felsner-Weil and Balko-Fulek-Kynčl reference). - Manfred Scheucher, Oct 20 2019
There is a relation to non-degenerate oriented matroids of rank 3 on n elements (see Folkman-Lawrence reference). - Manfred Scheucher, Feb 09 2022, based on comment by Matthew J. Samuel, Jan 19 2013
Also the number of tilings of the 2-dimensional zonotope constructed from D vectors. The zonotope Z(D,d) is the projection of the D-dimensional hypercube onto the d-dimensional space and the tiles are the projections of the d-dimensional faces of the hypercube. Here d=2 and D varies.
Also the number of simple arrangements of n pseudolines in the Euclidean plane. - Manfred Scheucher, Jun 22 2023

			This is a wiring diagram, one sample of the 62 objects that are counted for n=5:
  1-1-1-1 4-4 5-5
         X   X
  2 3-3 4 1 5 4-4
   X   X   X
  3 2 4 3 5 1 3-3
     X   X   X
  4-4 2 5 3-3 1 2
       X       X
  5-5-5 2-2-2-2 1
Each X denotes a comparator that exchanges the two incoming strands from the left. The whole network has n*(n-1)/2 such comparators and exchanges the order 12345 at the left edge into the reverse order 54321 at the right edge. It is also a pseudoline arrangement consisting of n x-monotone curves (from left to right), which pairwise cross exactly once.

Shin-ichi Minato, Counting by ZDD, Encyclopedia of Algorithms, 2014, pp. 1-6.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. Armstrong, The sorting order on a Coxeter group, Journal of Combinatorial Theory 116 (2009), no. 8, 1285-1305.
M. Balko, R. Fulek, and J. Kynčl, Crossing Numbers and Combinatorial Characterization of Monotone Drawings of K_n, Discrete & Computational Geometry, Volume 53, Issue 1, 2015, Pages 107-143.
Helena Bergold, Stefan Felsner, and Manfred Scheucher, Extendability of higher dimensional signotopes, Proc. 38th Eur. Wksp. Comp. Geom. (EuroCG), 2022. See also arXiv:2303.04079 [math.CO], 2023.
Herman Chau, On Enumerating Higher Bruhat Orders Through Deletion and Contraction, arXiv:2412.10532 [math.CO], 2024. See p. 2.
Yunhyung Cho, Jang Soo Kim, and Eunjeong Lee, Enumerate of Gelfand-Cetlin type reduced words, arXiv:2009.06906 [math.CO], 2020. Mentions this sequence.
Fernando Cortés Kühnast, Justin Dallant, Stefan Felsner, and Manfred Scheucher, An Improved Lower Bound on the Number of Pseudoline Arrangements, arXiv:2402.13107 [math.CO], 2024. See p. 2.
H. Denoncourt, D. C. Ernst, and D. Story, On the number of commutation classes of the longest element in the symmetric group, arXiv:1602.08328 [math.HO], 2016.
Adrian Dumitrescu and Ritankar Mandal, New Lower Bounds for the Number of Pseudoline Arrangements, arXiv:1809.03619 [math.CO], 2018.
Stefan Felsner, On the number of arrangements of pseudolines, Proceedings of the twelfth annual symposium on Computational geometry. ACM, 1996. Also Discrete and Computational Geometry, 18 (1997),257-267. Gives 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
S. Felsner and P. Valtr, Coding and Counting Arrangements of Pseudolines, Discrete and Computational Geometry, 46(3) (2011), 405-416.
S. Felsner and H. Weil, Sweeps, arrangements and signotopes, Discrete Applied Mathematics Volume 109, Issues 1-2, 2001, Pages 67-94.
J. Folkman and J. Lawrence, Oriented matroids, Journal of Combinatorial Theory, Series B 25 (1978), no. 2, 199-236.
Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry, CRC Press, 2017, see Table 5.6.1. [General reference for 2017 edition of the Handbook] - _N. J. A. Sloane_, Nov 14 2023
M. J. Hay, J. Schiff, and N. J. Fisch, Available free energy under local phase space diffusion, arXiv preprint arXiv:1604.08573 [math-ph], 2016, see Footnote 27.
J. Kawahara, T. Saitoh, R. Yoshinaka, and S. Minato, Counting Primitive Sorting Networks by PiDDs, Hokkaido University, Division of Computer Science, TCS Technical Reports, TCS-TR-A-11-54, Oct. 2011.
D. E. Knuth, Axioms and Hulls, Lect. Notes Comp. Sci., Vol. 606 (1992) p. 35. [From _R. J. Mathar_, Apr 02 2009]
Ricardo Mamede, The commutation classes of a permutation, 14th Comb. Days, Univ. Coimbra (Portugal, 2024). See slide 10.
S. Minato, Techniques of BDD/ZDD: Brief History and Recent Activity, IEICE Transactions on Information and Systems, Vol. E96-D, No. 7, pp.1419-1429.
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.
Matthew J. Samuel, Word posets, complexity, and Coxeter groups, arXiv:1101.4655 [math.CO], 2011.
M. J. Samuel, Word posets, with applications to Coxeter groups, arXiv preprint arXiv:1108.3638 [cs.DM], 2011.
Manfred Scheucher, C++ program for enumeration
B. E. Tenner, Tiling-based models of perimeter and area, arXiv:1811.00082 [math.CO], 2018.
M. Widom, N. Destainville, R. Mosseri, and F. Bailly, Two-dimensional random tilings of large codimension, Proceedings of the 7th International Conference on Quasicrystals (ICQ7, Stuttgart), arXiv:cond-mat/9912275 [cond-mat.stat-mech], 1999.
M. Widom, N. Destainville, R. Mosseri, and F. Bailly, Two-dimensional random tilings of large codimension, Materials Science and Engineering: A, Volumes 294-296, 15 December 2000, Pages 409-412.
Katsuhisa Yamanaka, Takashi Horiyama, Takeaki Uno, and Kunihiro Wasa, Ladder-Lottery Realization, 30th Canadian Conference on Computational Geometry (CCCG 2018) Winnipeg.
K. Yamanaka, S. Nakano, Y. Matsui, R. Uehara, and K. Nakada, Efficient enumeration of all ladder lotteries and its application, Theoretical Computer Science, Vol. 411, pp. 1714-1722, 2010.
G. M. Ziegler, Higher Bruhat Orders and Cyclic Hyperplane Arrangements, Topology, Volume 32, 1993.
Index entries for sequences related to sorting.

Cf. A006246. See A005118 for primitive sorting networks with exactly one comparator ("X") per column. See A060596-A060601 for higher dimensions. Cf. A006247.

# classes: Wrapper for computing number of commutation classes;
#   pass a permutation as a list
# Returns number of commutation classes of reduced words
# Longest element is of the form [n, n-1, ..., 1] (see Comments)
classes:=proc(perm) option remember:
    RETURN(classesRecurse(Array(perm), 0, 1)):
end:
#classesRecurse: Recursive procedure for computing number of commutation classes
classesRecurse:=proc(perm, spot, negs) local swaps, i, sums, c, doneany:
    sums:=0:
    doneany:=0:
    for i from spot to ArrayNumElems(perm)-2 do
        if perm[i+1]>perm[i+2] then
            swaps:=perm[i+1]:
            perm[i+1]:=perm[i+2]:
            perm[i+2]:=swaps:
            c:=classes(convert(perm, `list`)):
            sums:=sums+negs*c+classesRecurse(perm, i+2, -negs):
            swaps:=perm[i+1]:
            perm[i+1]:=perm[i+2]:
            perm[i+2]:=swaps:
            doneany:=1:
        end:
    end:
    if spot=0 and doneany=0 then RETURN(1):
    else RETURN(sums):
    end:
end:
seq(classes([seq(n+1-i, i = 1 .. n)]), n = 1 .. 9)
# Matthew J. Samuel, Jan 23 2011, Jan 26 2011

classes[perm_List] := classes[perm] = classesRecurse[perm, 0, 1];
classesRecurse[perm_List, spot_, negs_] := Module[{swaps, i, Sums, c, doneany, prm = perm}, Sums = 0; doneany = 0; For[i = spot, i <= Length[prm]-2, i++, If[prm[[i+1]] > prm[[i+2]], swaps = prm[[i+1]]; prm[[i+1]] = prm[[i+2]]; prm[[i+2]] = swaps; c = classes[prm]; Sums = Sums + negs*c + classesRecurse[prm, i+2, -negs]; swaps = prm[[i+1]]; prm[[i+1]] = prm[[i+2]]; prm[[i+2]] = swaps; doneany = 1]]; If[spot == 0 && doneany == 0, Return[1], Return[Sums]]];
a[n_] := a[n] = classes[Range[n] // Reverse];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 9}] (* Jean-François Alcover, May 09 2017, translated from Maple *)

Felsner and Valtr show that 0.1887 <= log_2(a(n))/n^2 <= 0.6571 for sufficiently large n. - Jeremy Tan, Nov 20 2017

Dumitrescu and Mandal improved the lower bound to 0.2083 <= log_2(a(n))/n^2 for sufficiently large n. - Manfred Scheucher, Sep 13 2021

More terms from Sebastien Veigneau (sv(AT)univ-mlv.fr), Jan 15 1997
a(10) confirmed by Katsuhisa Yamanaka(yamanaka(AT)hol.is.uec.ac.jp), May 06 2009. This value was also confirmed by Takashi Horiyama of Saitama Univ.
a(11) from Katsuhisa Yamanaka(yamanaka(AT)hol.is.uec.ac.jp), May 06 2009
Reference with formula that the Maple program implements added and a(11) verified by Matthew J. Samuel, Jan 25 2011
Removed invalid comment concerning the denominators of the indicated polynomials; added a(12). - Matthew J. Samuel, Jan 30 2011
a(13) from Toshiki Saitoh, Oct 17 2011
a(14) and a(15) from Yuma Tanaka, Aug 20 2013
a(16) by Günter Rote, Dec 01 2021

A014540 Rectilinear crossing number of complete graph on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 9, 19, 36, 62, 102, 153, 229, 324, 447, 603, 798, 1029, 1318, 1657, 2055, 2528, 3077, 3699, 4430, 5250, 6180

Offset: 1

Eric W. Weisstein

The values a(19) and a(21) were obtained by Aichholzer et al. in 2006. The value a(18) is claimed by the Rectilinear Crossing Number project after months of distributed computing. This was confirmed by Abrego et al., they also found the values a(20) and a(22) to a(27). The next unknown entry, a(28), is either 7233 or 7234. - Bernardo M. Abrego (bernardo.abrego(AT)csun.edu), May 05 2008

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.18, p. 532.
M. Gardner, Crossing Numbers. Ch. 11 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986.
C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.

B. M. Abrego, S. Fernandez-Merchant, J. Leaños, and G. Salazar, The maximum number of halving lines and the rectilinear crossing number of K_n for n <= 27, Electronic Notes in Discrete Mathematics, 30 (2008), 261-266.
O. Aichholzer, Crossing number project.
O. Aichholzer, F. Aurenhammer, and H. Krasser, Progress on rectilinear crossing numbers. [Broken link]
O. Aichholzer, F. Aurenhammer, and H. Krasser, Progress on rectilinear crossing numbers, Technical report, IGI-TU Graz, Austria, 2001.
O. Aichholzer, F. Aurenhammer, and H. Krasser, On the Rectilinear Crossing Number. [Broken link]
O. Aichholzer, J. Garcia, D. Orden, and P. Ramos, New lower bounds for the number of <= k-edges and the rectilinear crossing number of K_n, Discrete & Computational Geometry 38 (2007), 1-14.
O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001. [Broken link]
D. Archdeacon, The rectilinear crossing number.
D. Bienstock and N. Dean, Bounds for rectilinear crossing numbers, J. Graph Theory 17 (1993) 333-348
A. Brodsky, S. Durocher, and E. Gethner, The Rectilinear Crossing Number of K_{10} is 62, The Electronic J. Combin, #R23, 2001.
A. Brodsky, S. Durocher, and E. Gethner, Toward the rectilinear crossing number of K_n: new drawings, upper bounds, and asymptotics, Discrete Math. 262 (2003), 59-77.
D. Garber, The Orchard crossing number of an abstract graph, arXiv:math/0303317 [math.CO], 2003-2009.
H. F. Jensen, An Upper Bound for the Rectilinear Crossing Number of the Complete Graph, J. Comb. Th. Ser. B 10, 212-216, 1971.
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Eric Weisstein's World of Mathematics, Rectilinear Crossing Number.
Eric Weisstein's World of Mathematics, Zarankiewicz's Conjecture.

Cf. A000241, A030179, A006247.

102 from Oswin Aichholzer (oswin.aichholzer(AT)tugraz.at), Aug 14 2001
153 from Hannes Krasser (hkrasser(AT)igi.tu-graz.ac.at), Sep 17 2001
More terms from Eric W. Weisstein, Nov 30 2006
More terms from Bernardo M. Abrego (bernardo.abrego(AT)csun.edu), May 05 2008

A063666 Euclidean order types: number of realizable order types of n points in the plane.

Original entry on oeis.org

1, 2, 3, 16, 135, 3315, 158817, 14309547, 2334512907

Offset: 3

Hannes Krasser (hkrasser(AT)igi.tu-graz.ac.at), Aug 22 2001

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

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.

O. Aichholzer, F. Aurenhammer and H. Krasser, Enumerating order types for small point sets with applications
O. Aichholzer, F. Aurenhammer and H. Krasser, Enumerating order types for small point sets with applications, Order 19(3):265-281, September 2002.
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
Stefan Felsner and J. E. Goodman, Pseudoline Arrangements. In: Toth, O'Rourke, Goodman (eds.) Handbook of Discrete and Computational Geometry, 3rd edn. CRC Press, 2018.
Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry, CRC Press, 2017, see Table 5.6.1. [General reference for 2017 edition of the Handbook] - _N. J. A. Sloane_, Nov 14 2023

Cf. A006247.

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

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

A006246 Number of simple arrangements of pseudolines in the projective plane with an oriented marked cell; number of oriented abstract order types of n points (distinguishing mirror-symmetric copies).

Original entry on oeis.org

1, 1, 1, 2, 3, 20, 242, 6405, 316835, 28627261, 4686329954, 1382939012729, 732955581630129

Offset: 1

N. J. A. Sloane

Previous name was: Number of primitive sorting networks on n elements.

M. H. Klin et al., "2D-configurations and clique-cyclic orientations of the graphs L(K_p)", pages 149-162 of Reports in Molecular Theory, Vol. 1, 1990.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. E. Knuth, Axioms and Hulls, Lect. Notes Comp. Sci., Vol. 606, Springer-Verlag, Berlin, Heidelberg, 1992, p.35, entry C_n.
Günter Rote, NumPSLA — An experimental research tool for pseudoline arrangements and order types, arXiv:2503.02336 [math.CO], 2025. See Table 5 on p. 18:17, last column.
Index entries for sequences related to sorting

Cf. A006245. See A006247 for abstract order types when mirror-symmetric copies are identified.

Definition corrected by Günter Rote, Dec 02 2021

a(12) and a(13) from Günter Rote, Mar 04 2025

A180503 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 Moebius strip.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 7, 16, 2, 13, 140, 1499, 11502, 3, 122, 5589, 245222, 9186477, 238834187

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 A180502 for isomorphism classes of all (not only simple) arrangements of n pseudolines and m double pseudolines in the Moebius strip.
See A180500 for isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the projective plane.
First diagonal gives A006247.

A276096 a(n) is the least number of empty convex pentagons ("convex 5-holes") spanned by a set of n points in the Euclidean plane in general position (i.e., no three points on a line).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 3, 6, 9, 11

Offset: 1

Manfred Scheucher, Aug 18 2016

The value a(10) = 1 was determined by Harborth, who also constructed a set of 9 points without convex 5-holes. The values a(11) = 2 and a(13) = 3 were determined by Dehnhardt. Aichholzer found point sets showing that a(14) <= 6 and a(15) <= 9, and the exact values a(13) = 3, a(14) = 6, and a(15) = 9 were determined in the Bachelor's thesis of Scheucher, supervised by Aichholzer and Hackl.

The value a(16) = 11 was determined using a ILP/SAT solver. For more information check out the link below with title "On 5-Holes". - Manfred Scheucher, Aug 18 2018

K. Dehnhardt, Leere konvexe Vielecke in ebenen Punktmengen, PhD thesis, TU Braunschweig, Germany, 1987, in German.

O. Aichholzer, M. Balko, T. Hackl, J. Kynčl, I. Parada, M. Scheucher, P. Valtr, and B. Vogtenhuber, A superlinear lower bound on the number of 5-holes, arXiv:1703.05253 [math.CO], 2017.
O. Aichholzer, R. Fabila-Monroy, T. Hackl, C. Huemer, A. Pilz, and B. Vogtenhuber, Lower bounds for the number of small convex k-holes, Computational Geometry: Theory and Applications, 47(5):605-613, 2014.
EuroGIGA - CRP ComPoSe, A set of 13 points with 3 convex 5-holes
EuroGIGA - CRP ComPoSe, A set of 14 points with 6 convex 5-holes
EuroGIGA - CRP ComPoSe, A set of 15 points with 9 convex 5-holes
EuroGIGA - CRP ComPoSe, A set of 16 points with 11 convex 5-holes
H. Harborth, Konvexe Fünfecke in ebenen Punktmengen, Elemente der Mathematik, 33:116-118, 1978, in German.
M. Scheucher, Counting Convex 5-Holes, Bachelor's thesis, Graz University of Technology, Austria, 2013, in German.
M. Scheucher, On 5-Holes.

Cf. A063541 and A063542 for convex 3- and 4-holes, respectively.

Cf. A006247 and A063666 for equivalence classes (w.r.t. orientation triples) of point sets in the plane.

From Manfred Scheucher, Mar 22 2017: (Start)
a(n) = Omega(n log^(4/5)(n)) and a(n) = O(n^2).
Conjecture: a(n) = Theta(n^2). (End)

a(16) from Manfred Scheucher, Mar 22 2017

cp's OEIS Frontend

A006245 Number of primitive sorting networks on n elements; also number of rhombic tilings of a 2n-gon.

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A014540 Rectilinear crossing number of complete graph on n nodes.

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A063666 Euclidean order types: number of realizable order types of n points in the plane.

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A018242 Number of projective order types.

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A006246 Number of simple arrangements of pseudolines in the projective plane with an oriented marked cell; number of oriented abstract order types of n points (distinguishing mirror-symmetric copies).

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A180503 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 Moebius strip.

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A276096 a(n) is the least number of empty convex pentagons ("convex 5-holes") spanned by a set of n points in the Euclidean plane in general position (i.e., no three points on a line).

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A325595 Number of symmetric Euclidean pseudo-order types: nondegenerate abstract order types of configurations of n points in the plane with a nontrivial automorphism.

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A325628 Number of mirror-symmetric Euclidean pseudo-order types: nondegenerate abstract order types of configurations of n points in the plane with a mirroring automorphism.

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