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

1, 2, 20, 7658, 12954016496, 10592917773063552232751878

Offset

8,2

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=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

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=8..13.

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.

Formula

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

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=8 of A060637.

Sequence in context: A224732 A055746 A258878 * A356689 A346564 A249646

Adjacent sequences: A060597 A060598 A060599 * A060601 A060602 A060603

Keyword

nonn,nice,hard,more

Author

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

Extensions

a(12) from Manfred Scheucher, Nov 30 2021

Edited by Manfred Scheucher, Mar 08 2022

a(13) from Manfred Scheucher, Aug 06 2023

Status

approved