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

1, 2, 16, 1646, 16930560, 665354510109750

Offset

6,2

Comments

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

References

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.

Links

Table of n, a(n) for n=6..11.

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.

Formula

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

Example

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.

Crossrefs

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

Column k=6 of A060637.

Sequence in context: A102103 A337070 A326974 * A091479 A016031 A331691

Adjacent sequences: A060594 A060595 A060596 * A060598 A060599 A060600

Keyword

nonn,nice

Author

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

Extensions

a(10) from Manfred Scheucher, Sep 21 2021

Edited by Manfred Scheucher, Mar 08 2022

a(11) from Manfred Scheucher, Jul 17 2023

Status

approved