A060595 -id:A060595 - cp's OEIS frontend

A060598 Number of tilings of the 7-dimensional zonotope constructed from D vectors.

1, 2, 18, 3564, 393454160, 24410990062379593896

Offset: 7

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=7 and D varies.

Also the number of signotopes of rank 8. 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=7 of A060637.

Asymptotics: a(n) = 2^(Theta(n^7)). 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^7} <= a(n) <= 2^{d n^7} is satisfied. - Manfred Scheucher, Sep 22 2021

a(11) from Manfred Scheucher, Sep 22 2021
Edited by Manfred Scheucher, Mar 08 2022
a(12) from Manfred Scheucher, Jul 17 2023

A060599 Number of tilings of the 5-dimensional zonotope constructed from D vectors.

Original entry on oeis.org

1, 2, 14, 752, 1000488, 183886016052, 58898534395717170440

Offset: 5

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=5 and D varies.

Also the number of signotopes of rank 6. 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=5 of A060637.

Asymptotics: a(n) = 2^(Theta(n^5)). 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^5} <= a(n) <= 2^{d n^5} is satisfied. - Manfred Scheucher, Sep 22 2021

a(9) from Manfred Scheucher, Sep 21 2021
a(10) from Manfred Scheucher, Oct 20 2021
Edited by Manfred Scheucher, Mar 08 2022
a(11) from Manfred Scheucher, Jul 17 2023

A060614 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=5 and D varies.

Original entry on oeis.org

0, 1, 14, 1664

Offset: 5

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 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.
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.
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.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A060616 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=6 and D varies.

Original entry on oeis.org

0, 1, 16, 4032

Offset: 6

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 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.
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.
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.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A060619 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=9 and D varies.

Original entry on oeis.org

0, 1, 22, 52224

Offset: 9

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 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.
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.
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.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A060621 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here the codimension D-d is equal to 3 and d varies.

Original entry on oeis.org

12, 36, 100, 264, 672, 1664, 4032, 9600, 22528, 52224

Offset: 0

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 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.
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.
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.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

Numbers so far satisfy a(n) = 2^n*(n^2+11n+24)/2. - Ralf Stephan, Apr 08 2004

Empirical g.f.: -4*(7*x^2-9*x+3) / (2*x-1)^3. - Colin Barker, Feb 20 2013

A060624 Number d-dimensional tilings of the unary zonotope Z(D,d). Here the codimension D-d is equal to 5 and d varies.

Original entry on oeis.org

32, 720, 24698

Offset: 0

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

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.
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.
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.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A119886 a(n) = 20a(n-2) - 64a(n-4).

Original entry on oeis.org

1, 59, 2416, 6230, 47680, 120824, 798976, 2017760, 12928000, 32622464, 207425536, 523312640, 3321118720, 8378415104, 53147140096, 134076293120, 850391203840, 2145307295744, 13606407110656, 34325263155200, 217703105167360, 549205596176384, 3483252048265216

Offset: 0

Roger L. Bagula, Aug 09 2006

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,20,0,-64).

Cf. A060595, A060638, A002409, A099193, A000937, A099195.

M = {{0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0}, {1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0}} v[1] = Table[Fibonacci[n], {n, 1, 16}] v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
(* Second program: *)
A = SparseArray[{{1, 8} -> 1, Band[{1, 4}] -> 1, Band[{1, 2}, {3, 4}] -> 1, Band[{5, 6}, {7, 8}] -> 1}, {8, 8}]; M = ArrayFlatten[{{A+Transpose[A], IdentityMatrix[8]}, {IdentityMatrix[8], A+Transpose[A]}}]; v[1] = Array[ Fibonacci, 16]; v[n_] := v[n] = M.v[n-1]; A119886 = Array[v, 50][[All, 1]] (* Jean-François Alcover, Feb 05 2017 *)
LinearRecurrence[{0,20,0,-64},{1,59,2416,6230,47680},30] (* Harvey P. Dale, Sep 06 2024 *)

Vec(-(576*x^4-5050*x^3-2396*x^2-59*x-1) / ((2*x-1)*(2*x+1)*(4*x-1)*(4*x+1)) + O(x^30)) \\ Colin Barker, Feb 05 2017

G.f.: -(576*x^4-5050*x^3-2396*x^2-59*x-1) / ((2*x-1)*(2*x+1)*(4*x-1)*(4*x+1)). - Colin Barker, Nov 17 2012

a(n) = 2^(n-4)*(-3266 + 585*(-2)^n + 258*(-1)^n + 2583*2^n) for n>0. - Colin Barker, Feb 05 2017

New name from Joerg Arndt, Feb 05 2017

A060600 Number of tilings of the 8-dimensional zonotope constructed from D vectors.

Original entry on oeis.org

1, 2, 20, 7658, 12954016496, 10592917773063552232751878

Offset: 8

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=8 and D varies.

Also the number of signotopes of rank 9. 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.
G. M. Ziegler, Higher Bruhat Orders and Cyclic Hyperplane Arrangements, Topology, Volume 32, 1993.

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

Column k=8 of A060637.

Asymptotics: a(n) = 2^(Theta(n^8)). 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^8} <= a(n) <= 2^{d n^8} is satisfied. - Manfred Scheucher, Sep 22 2021

a(12) from Manfred Scheucher, Nov 30 2021
Edited by Manfred Scheucher, Mar 08 2022
a(13) from Manfred Scheucher, Aug 06 2023

A060617 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=7 and D varies.

Original entry on oeis.org

0, 1, 18, 9600

Offset: 7

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 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.
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.
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.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

cp's OEIS Frontend

A060598 Number of tilings of the 7-dimensional zonotope constructed from D vectors.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A060599 Number of tilings of the 5-dimensional zonotope constructed from D vectors.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A060614 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=5 and D varies.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A060616 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=6 and D varies.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A060619 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=9 and D varies.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A060621 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here the codimension D-d is equal to 3 and d varies.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Formula

A060624 Number d-dimensional tilings of the unary zonotope Z(D,d). Here the codimension D-d is equal to 5 and d varies.

Views

Author

Keywords

References

Links

Crossrefs

A119886 a(n) = 20*a(n-2) - 64*a(n-4).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A060600 Number of tilings of the 8-dimensional zonotope constructed from D vectors.

Views

Author

Keywords

Comments

Examples

A119886 a(n) = 20a(n-2) - 64a(n-4).