A060595 -id:A060595 - cp's OEIS frontend

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

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

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

A351383 Number of tilings of the d-dimensional zonotope constructed from d+4 vectors.

Original entry on oeis.org

16, 120, 908, 7686, 78032, 1000488, 16930560, 393454160, 12954016496, 613773463394

Offset: 0

Manfred Scheucher, Feb 09 2022

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 = 4 and d >= 0.

Also the number of signotopes on r+3 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).

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.

A diagonal of A060637.

Cf. A006245 (two-dimensional tilings), A060595-A060602, A351384.

A351384 Number of tilings of the d-dimensional zonotope constructed from d+5 vectors.

Original entry on oeis.org

32, 720, 24698, 1681104, 295118262, 183886016052

Offset: 0

Manfred Scheucher, Feb 09 2022

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 = 5 and d >= 0.

Also the number of signotopes on r+4 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).

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.

A diagonal of A060637.

Cf. A006245 (two-dimensional tilings), A060595-A060601, A060602, A351383.

A060570 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=2 and D varies.

Original entry on oeis.org

0, 1, 8, 100, 2144, 80360

Offset: 2

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, arXiv:math/0008022 [math.CO], 2000.

Cf. A001286 (case where d=1), A006245 (number of 2-tilings). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A060608 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=3 and D varies.

Original entry on oeis.org

0, 1, 10, 264, 22624

Offset: 3

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, arXiv:math/0008022 [math.CO], 2000.

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

A060612 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=4 and D varies.

Original entry on oeis.org

0, 1, 12, 672

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.

A060597 Number of tilings of the 6-dimensional zonotope constructed from D vectors.

Original entry on oeis.org

1, 2, 16, 1646, 16930560, 665354510109750

Offset: 6

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=6 and D varies.

Also the number of signotopes of rank 7. 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=6 of A060637.

Asymptotics: a(n) = 2^(Theta(n^6)). 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^6} <= a(n) <= 2^{d n^6} is satisfied. - Manfred Scheucher, Sep 22 2021

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

cp's OEIS Frontend

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

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

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A351383 Number of tilings of the d-dimensional zonotope constructed from d+4 vectors.

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A351384 Number of tilings of the d-dimensional zonotope constructed from d+5 vectors.

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A060570 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=2 and D varies.

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A060608 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=3 and D varies.

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A060612 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=4 and D varies.

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