A006245 -id:A006245 - cp's OEIS frontend

A060595 Number of tilings of the 3-dimensional zonotope constructed from D vectors.

1, 2, 10, 148, 7686, 1681104, 1881850464, 13227777493060

Offset: 3

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 12 2001

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=3 and D varies.

Also the number of signotopes of rank 4, i.e., mappings X:{{1..n} choose 4}->{+,-} such that for any four indices a < b < c < d < e, the sequence X(a,b,c,d), X(a,b,c,e), X(a,b,d,e), X(a,c,d,e), X(b,c,d,e), changes its sign at most once (see Felsner-Weil reference). - Manfred Scheucher, Sep 13 2021

			Z(3,3) is simply a cube and the only possible tile is Z(3,3) itself, therefore the first term of the series is 1. It is well known that there are always two d-tilings of Z(d+1,d), therefore the second term is 2. More examples are available on my web page.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G.M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999
V. Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

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.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, arXiv:cond-mat/0004145 [cond-mat.stat-mech], 2000.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
S. Felsner and H. Weil, Sweeps, arrangements and signotopes, Discrete Applied Mathematics, Volume 109, Issues 1-2, 2001, Pages 67-94.
M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices, arXiv:math/0008022 [math.CO], 2000.
J. A. Olarte and F. Santos, Hypersimplicial subdivisions, arXiv:1906.05764 [math.CO], 2019.
Manfred Scheucher, C program for enumeration
G. M. Ziegler, Higher Bruhat Orders and Cyclic Hyperplane Arrangements, Topology, Volume 32, 1993.

Cf. A006245 (two-dimensional tilings), A060596-A060602.

Column k=3 of A060637.

Asymptotics: a(n) = 2^(Theta(n^3)). 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^3} <= a(n) <= 2^{d n^3} is satisfied. - Manfred Scheucher, Sep 22 2021

a(8)-a(9) from Manfred Scheucher, Sep 13 2021
Edited by Manfred Scheucher, Mar 08 2022
a(10) from Manfred Scheucher, Jul 17 2023

A006248 Number of projective pseudo order types: simple arrangements of pseudo-lines in the projective plane.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 11, 135, 4382, 312356, 41848591, 10320613331

Offset: 1

N. J. A. Sloane

J. Bokowski, personal communication.
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 102.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Bokowski & N. J. A. Sloane, Emails, June 1994
F. Cortés Kühnast, J. Dallant, S. Felsner, and M. Scheucher, An Improved Lower Bound on the Number of Pseudoline Arrangements
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 J. E. Goodman, Pseudoline Arrangements. In: Toth, O'Rourke, Goodman (eds.) Handbook of Discrete and Computational Geometry, 3rd edn. CRC Press, 2018.
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.
L. 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.
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. - From _N. J. A. Sloane_, Feb 16 2013
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
D. E. Knuth, Axioms and Hulls, Lect. Notes Comp. Sci., Vol. 606, Springer-Verlag, Berlin, Heidelberg, 1992, p.35, entry E_n.
Index entries for sequences related to sorting

Cf. A006245, A006246, A018242, A063666. A diagonal of A063851.

Asymptotics: 2^{Cn^2} <= a(n) <= 2^{Dn^2} for every n >= N, where N,C,D are constants with 0.1887Manfred Scheucher, Apr 10 2025 on personal communication with Günter Rote.]

a(11) from Franz Aurenhammer (auren(AT)igi.tu-graz.ac.at), Feb 05 2002
a(12) from Manfred Scheucher and Günter Rote, Sep 07 2019
Definition corrected by Günter Rote, Dec 01 2021

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

A060637 Triangle T(n,k) (0 <= k <= n) giving number of tilings of the k-dimensional zonotope constructed from n vectors.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 6, 2, 1, 16, 24, 8, 2, 1, 32, 120, 62, 10, 2, 1, 64, 720, 908, 148, 12, 2, 1, 128, 5040, 24698, 7686, 338, 14, 2, 1, 256, 40320, 1232944

Offset: 0

N. J. A. Sloane, Apr 16 2001

The zonotope Z(n,k) is the projection of the n-dimensional hypercube onto the k-dimensional space and the tiles are the projections of the k-dimensional faces of the hypercube.

T(n,k) is also the number of signotopes on n elements of rank r=k+1. A signotope on n elements of rank r is a mapping X:{{1..n} choose r}->{+,-} such that for any r+1 indices I={i_0,...,i_r} with i_0 < i_1 < ... < i_r, the sequence X(I-i_0), X(I-i_1), ..., X(I-i_r) changes its sign at most once (see Felsner-Weil reference). - Manfred Scheucher, Feb 09 2022

			Triangle T(n,k) begins:
    1;
    2,    1;
    4,    2,     1;
    8,    6,     2,    1;
   16,   24,     8,    2,   1;
   32,  120,    62,   10,   2,  1;
   64,  720,   908,  148,  12,  2, 1;
  128, 5040, 24698, 7686, 338, 14, 2, 1;
  ...

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, arXiv:cond-mat/0004145 [cond-mat.stat-mech], 2000.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
S. Felsner and H. Weil, Sweeps, arrangements and signotopes, Discrete Applied Mathematics, Volume 109, Issues 1-2, 2001, Pages 67-94.
M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices, arXiv:math/0008022 [math.CO], 2000.

Diagonals give A000079, A000142, A006245, A060595-A060602, A351383, A351384. Cf. A060638.

Edited by Manfred Scheucher, Mar 08 2022

A060602 Number of tilings of the d-dimensional zonotope constructed from d+3 vectors.

Original entry on oeis.org

8, 24, 62, 148, 338, 752, 1646, 3564, 7658, 16360, 34790, 73700, 155618, 327648, 688094, 1441756, 3014618, 6291416, 13107158, 27262932, 56623058, 117440464, 243269582, 503316428, 1040187338, 2147483592, 4429184966

Offset: 0

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 12 2001

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 the codimension, i.e., D-d, is constant = 3 and d >= 0.

Also the number of signotopes on r+2 elements of rank r. A signotope on n elements of rank r is a mapping X:{{1..n} choose r}->{+,-} such that for any r+1 indices I={i_0,...,i_r} with i_0 < i_1 < ... < i_r, the sequence X(I-i_0), X(I-i_1), ..., X(I-i_r) changes its sign at most once (see Felsner-Weil reference). - Manfred Scheucher, Feb 09 2022

			For any Z(D,d), the number of codimension 0 tilings is always 1, with codimension 1 it is 2, with codimension 2 it is 2.D.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, arXiv:cond-mat/0004145 [cond-mat.stat-mech], 2000.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
S. Felsner and H. Weil, Sweeps, arrangements and signotopes, Discrete Applied Mathematics, Volume 109, Issues 1-2, 2001, Pages 67-94.
M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices, arXiv:math/0008022 [math.CO], 2000.
G. M. Ziegler, Higher Bruhat Orders and Cyclic Hyperplane Arrangements, Topology, Volume 32, 1993.
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A006245 (two-dimensional tilings), A060595-A060601. A diagonal of A060637. See also A351383 and A351384 for other diagonals.

Cf. A133546.

LinearRecurrence[{6,-13,12,-4},{8,24,62,148},30] (* Harvey P. Dale, Oct 13 2023 *)

print([2**(n + 1)*(n + 7) - 2*n - 6 for n in range(100)])

Conjectures from Colin Barker, Feb 20 2013: (Start)
a(n) = 2*(-3+7*2^n+(-1+2^n)*n).
G.f.: -2*(4*x^3-11*x^2+12*x-4) / ((x-1)^2*(2*x-1)^2). (End)
The above conjectures are correct; see Proposition 7.1 in Ziegler's article. - Manfred Scheucher, Feb 09 2022
a(n) = 2 * A133546(n+2). - Alois P. Heinz, Feb 11 2022

Edited by Manfred Scheucher, Mar 08 2022

A086903 a(n) = 8*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 8.

Original entry on oeis.org

2, 8, 62, 488, 3842, 30248, 238142, 1874888, 14760962, 116212808, 914941502, 7203319208, 56711612162, 446489578088, 3515205012542, 27675150522248, 217885999165442, 1715412842801288, 13505416743244862, 106327921103157608

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 21 2003

a(n+1)/a(n) converges to (4+sqrt(15)) = 7.872983... a(0)/a(1)=2/8; a(1)/a(2)=8/62; a(2)/a(3)=62/488; a(3)/a(4)=488/3842; ... etc. Lim a(n)/a(n+1) as n approaches infinity = 0.127016... = 1/(4+sqrt(15)) = (4-sqrt(15)).
Twice A001091. - John W. Layman, Sep 25 2003
Except for the first term, positive values of x (or y) satisfying x^2 - 8xy + y^2 + 60 = 0. - Colin Barker, Feb 13 2014

			a(4) = 3842 = 8*a(3) - a(2) = 8*488 - 62 = (4+sqrt(15))^4 + (4-sqrt(15))^4 = 3841.9997397 + 0.0002603 = 3842.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
P. Bala, Some simple continued fraction expansions for an infinite product, Part 1
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (8,-1).

Cf. A086594, A058316, A006245, A009271, A005248, A174502.

I:=[2,8]; [n le 2 select I[n] else 8*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 15 2014

a[0] = 2; a[1] = 8; a[n_] := 8a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 19}] (* Robert G. Wilson v, Jan 30 2004 *)
CoefficientList[Series[(2 - 8 x)/(1 - 8 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 15 2014 *)
LinearRecurrence[{8,-1},{2,8},30] (* Harvey P. Dale, Jan 18 2015 *)

[lucas_number2(n,8,1) for n in range(27)] # Zerinvary Lajos, Jun 25 2008

a(n) = (4+sqrt(15))^n + (4-sqrt(15))^n.
G.f.: (2-8*x)/(1-8*x+x^2). [Philippe Deléham, Nov 02 2008]
From Peter Bala, Jan 06 2013: (Start)
Let F(x) = product {n = 0..inf} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let alpha = 4 - sqrt(15). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.12474 84992 41370 33639 ... = 2 + 1/(8 + 1/(62 + 1/(488 + ...))).  Cf. A174502 and A005248.
Also F(-alpha) = 0.87474 74663 84045 35032 ... has the continued fraction representation 1 - 1/(8 - 1/(62 - 1/(488 - ...))) and the simple continued fraction expansion 1/(1 + 1/((8-2) + 1/(1 + 1/((62-2) + 1/(1 + 1/((488-2) + 1/(1 + ...))))))).
F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((8^2-4) + 1/(1 + 1/((62^2-4) + 1/(1 + 1/((488^2-4) + 1/(1 + ...))))))).
(End)

A145662 a(n) = numerator of polynomial of genus 1 and level n for m = 5 = A[1,n](5).

Original entry on oeis.org

0, 5, 55, 835, 8365, 41837, 209195, 7321885, 73218955, 1098284605, 5491423277, 302028282755, 1510141416085, 98159192073245, 490795960391965, 2453979801983849, 24539798019883535, 2085882831690821195

Offset: 1

Artur Jasinski, Oct 16 2008

For numerator of polynomial of genus 1 and level n for m = 1 see A001008
Definition: The polynomial A[1,2n+1](m) = A[genus 1,level n] is here defined as
Sum_{d=1..n-1} m^(n-d)/d.
Few first A[1,n](m):
n=1: A[1,1](m)= 0;
n=2: A[1,2](m)= m;
n=3: A[1,3](m)= m/2 + m^2;
n=4: A[1,4](m)= m/4 + m^2/3 + m^3/2 + m^4.
General formula which uses these polynomials is:
(1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] = Sum_{x>=0} m^(-x)/(x+n) = m^n*arctanh((2m-1)/(2m^2-2m+1)) - A[1,n](m) = m^n*log(m/(m-1)) - A[1,n](m).
The sequence of denominators is ?, 1, 2, 6, 12, 12, 12, 84, ... - Matthew J. Samuel, Jan 30 2011

Cf. A145609-A145640, A145656, A145658, A145660, A145664, A145666.

Cf. A006245.

A145662 := proc(n) add( 5^(n-d)/d,d=1..n-1) ; numer(%) ; end proc: # R. J. Mathar, Feb 01 2011

m = 5; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa

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

A060596 Number of tilings of the 4-dimensional zonotope constructed from D vectors.

Original entry on oeis.org

1, 2, 12, 338, 78032, 295118262, 42185916295296

Offset: 4

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 12 2001

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=4 and D varies.

Also the number of signotopes of rank 5. A signotope of rank r is a mapping X:{{1..n} choose r}->{+,-} such that for any r+1 indices I={i_0,...,i_r} with i_0 < i_1 < ... < i_r, the sequence X(I-i_0), X(I-i_1), ..., X(I-i_r) changes its sign at most once (see Felsner-Weil reference). - Manfred Scheucher, Feb 09 2022

			For any d, the only possible tile for Z(d,d) is Z(d,d) itself, therefore the first term of the series is 1. It is well known that there are always two d-tilings of Z(d+1,d), therefore the second term is 2. More examples are available on my web page.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G.M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46 Second Edition, Cambridge University Press, 1999.
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, arXiv:cond-mat/0004145 [cond-mat.stat-mech], 2000.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
S. Felsner and H. Weil, Sweeps, arrangements and signotopes, Discrete Applied Mathematics, Volume 109, Issues 1-2, 2001, Pages 67-94.
M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices, arXiv:math/0008022 [math.CO], 2000.
Manfred Scheucher, C program for enumeration.
G. M. Ziegler, Higher Bruhat Orders and Cyclic Hyperplane Arrangements, Topology, Volume 32, 1993.

Cf. A006245 (two-dimensional tilings), A060595-A060602.

Column k=4 of A060637.

Asymptotics: a(n) = 2^(Theta(n^4)). 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^4} <= a(n) <= 2^{d n^4} is satisfied. - Manfred Scheucher, Sep 22 2021

a(8)-a(9) from Manfred Scheucher, Sep 21 2021
Edited by Manfred Scheucher, Mar 08 2022
a(10) from Manfred Scheucher, Jul 17 2023

A060601 Number of tilings of the 9-dimensional zonotope constructed from D vectors.

Original entry on oeis.org

1, 2, 22, 16360, 613773463394

Offset: 9

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 12 2001

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=9 and D varies.

Also the number of signotopes of rank 10. A signotope of rank r is a mapping X:{{1..n} choose r}->{+,-} such that for any r+1 indices I={i_0,...,i_r} with i_0 < i_1 < ... < i_r, the sequence X(I-i_0), X(I-i_1), ..., X(I-i_r) changes its sign at most once (see Felsner-Weil reference). - Manfred Scheucher, Feb 09 2022

			For any d, the only possible tile for Z(d,d) is Z(d,d) itself, therefore the first term of the series is 1. It is well known that there are always two d-tilings of Z(d+1,d), therefore the second term is 2. More examples are available on my web page.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G.M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46 Second Edition, Cambridge University Press, 1999.
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

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.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, arXiv:cond-mat/0004145 [cond-mat.stat-mech], 2000.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
S. Felsner and H. Weil, Sweeps, arrangements and signotopes, Discrete Applied Mathematics, Volume 109, Issues 1-2, 2001, Pages 67-94.
M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices, arXiv:math/0008022 [math.CO], 2000.
G. M. Ziegler, Higher Bruhat Orders and Cyclic Hyperplane Arrangements, Topology, Volume 32, 1993.

Cf. A006245 (two-dimensional tilings), A060595-A060602.

Column k=9 of A060637.

Asymptotics: a(n) = 2^(Theta(n^9)). 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^9} <= a(n) <= 2^{d n^9} is satisfied. - Manfred Scheucher, Sep 22 2021

a(13) from Manfred Scheucher, Mar 07 2022

Edited by Manfred Scheucher, Mar 08 2022

cp's OEIS Frontend

A060595 Number of tilings of the 3-dimensional zonotope constructed from D vectors.

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A006248 Number of projective pseudo order types: simple arrangements of pseudo-lines in the projective plane.

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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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A060637 Triangle T(n,k) (0 <= k <= n) giving number of tilings of the k-dimensional zonotope constructed from n vectors.

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A060602 Number of tilings of the d-dimensional zonotope constructed from d+3 vectors.

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A086903 a(n) = 8*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 8.

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A145662 a(n) = numerator of polynomial of genus 1 and level n for m = 5 = A[1,n](5).

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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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