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
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
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