A006245 -id:A006245 - cp's OEIS frontend

A328377 a(n) is the number of "generalized signotopes", 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 twice (equivalently +-+- and -+-+ are forbidden).

2, 14, 544, 173128, 630988832, 35355434970848

Offset: 3

Manfred Scheucher, Oct 14 2019

Clearly a generalization of "signotopes" (cf. A006245), 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).
Also a generalization of "simple topological drawings" (a.k.a. good drawings, cf. A276109), i.e., non-isomorphic drawings of the complete graph K_n such that any two edges intersect at most once. In a simple topological drawings, each three vertices a < b < c determine a triangle which is either oriented clockwise or counterclockwise -- this clearly motivates the mapping X. It can be checked that in any simple topological drawing of K_4, the sequence X(a,b,c), X(a,b,d), X(a,c,d), X(b,c,d) changes its sign at most twice.
Also known as "Interior triple systems", see Knuth's book.

D. Knuth, Axioms and Hulls, Springer, 1992, 9-11.

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.
H. Bergold, S. Felsner, M. Scheucher, F. Schröder, and R. Steiner, Topological Drawings meet Classical Theorems from Convex Geometry, Discrete & Computational Geometry, Springer, 2022.
S. Felsner and H. Weil, Sweeps, arrangements and signotopes, Discrete Applied Mathematics, Volume 109, Issues 1-2, 2001, Pages 67-94.
Manfred Scheucher, C-program for computing the first terms

Cf. A006245, A329980.

a(8) from Robert Lauff and Manfred Scheucher, Nov 04 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.

A320944 a(n) is the number of perimeter tiles among all rhombic tilings of the 2n-gon.

Original entry on oeis.org

1, 6, 24, 200, 3216

Offset: 2

Bridget Tenner, Oct 24 2018

Tiles and polygons all have sides of length 1, as in A006245.

			a(4) = 24 because among the eight rhombic tilings of the octagon, 24 perimeter rhombi appear.

B. E. Tenner, Tiling-based models of perimeter and area, arXiv:1811.00082 [math.CO], 2018.

Cf. A320945, A320946, A006245.

a(n) = 2*n*A320945(n) for n > 2.

A320945 a(n) is the number of top-perimeter tiles among all rhombic tilings of the 2n-gon.

Original entry on oeis.org

1, 1, 3, 20, 268

Offset: 2

Bridget Tenner, Oct 24 2018

Also the number of bottom-perimeter tiles among all rhombic tilings of the 2n-gon. Tiles and polygons all have sides of length 1, as in A006245.

			a(4) = 3 because among the eight rhombic tilings of the octagon, 3 top-perimeter rhombi appear.

B. E. Tenner, Tiling-based models of perimeter and area, arXiv:1811.00082 [math.CO], 2018.

Cf. A320944, A320946, A006245.

A320946 a(n) is the number of left-perimeter tiles among all rhombic tilings of the 2n-gon.

Original entry on oeis.org

1, 2, 9, 80, 1340

Offset: 2

Bridget Tenner, Oct 24 2018

Also the number of right-perimeter tiles among all rhombic tilings of the 2n-gon. Tiles and polygons all have sides of length 1, as in A006245.

			a(4) = 9 because among the eight rhombic tilings of the octagon, 9 left-perimeter rhombi appear.

B. E. Tenner, Tiling-based models of perimeter and area, arXiv:1811.00082 [math.CO], 2018.

Cf. A320944, A320945, A006245.

a(n) = (n-1) * A320945(n).

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

A328377 a(n) is the number of "generalized signotopes", 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 twice (equivalently +-+- and -+-+ are forbidden).

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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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A320944 a(n) is the number of perimeter tiles among all rhombic tilings of the 2n-gon.

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A320945 a(n) is the number of top-perimeter tiles among all rhombic tilings of the 2n-gon.

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A320946 a(n) is the number of left-perimeter tiles among all rhombic tilings of the 2n-gon.

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

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

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