A060595 Number of tilings of the 3-dimensional zonotope constructed from D vectors.

1, 2, 10, 148, 7686, 1681104, 1881850464, 13227777493060

Offset

3,2

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=3 and D varies.

Also the number of signotopes of rank 4, i.e., mappings X:{{1..n} choose 4}->{+,-} such that for any four indices a < b < c < d < e, the sequence X(a,b,c,d), X(a,b,c,e), X(a,b,d,e), X(a,c,d,e), X(b,c,d,e), changes its sign at most once (see Felsner-Weil reference). - Manfred Scheucher, Sep 13 2021

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

V. 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=3..10.

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.

J. A. Olarte and F. Santos, Hypersimplicial subdivisions, arXiv:1906.05764 [math.CO], 2019.

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^3)). 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^3} <= a(n) <= 2^{d n^3} is satisfied. - Manfred Scheucher, Sep 22 2021

Example

Z(3,3) is simply a cube and the only possible tile is Z(3,3) 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), A060596-A060602.

Column k=3 of A060637.

Sequence in context: A317075 A295207 A213457 * A303440 A086619 A294373

Adjacent sequences: A060592 A060593 A060594 * A060596 A060597 A060598

Keyword

nonn,nice

Author

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

Extensions

a(8)-a(9) from Manfred Scheucher, Sep 13 2021

Edited by Manfred Scheucher, Mar 08 2022

a(10) from Manfred Scheucher, Jul 17 2023

Status

approved