

A060595


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


24




OFFSET

3,2


COMMENTS

The zonotope Z(D,d) is the projection of the Ddimensional hypercube onto the ddimensional space and the tiles are the projections of the ddimensional 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 FelsnerWeil 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, 19961997), 293336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.


LINKS

Table of n, a(n) for n=3..9.
N. Destainville, R. Mosseri and F. Bailly, Fixedboundary octagonal random tilings: a combinatorial approach, arXiv:condmat/0004145 [condmat.statmech], 2000.
N. Destainville, R. Mosseri and F. Bailly, Fixedboundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 12, 147190.
S. Felsner and H. Weil, Sweeps, arrangements and signotopes, Discrete Applied Mathematics, Volume 109, Issues 12, 2001, Pages 6794.
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 BachmannLandau 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 dtilings of Z(d+1,d), therefore the second term is 2. More examples are available on my web page.


CROSSREFS

Cf. A006245 (twodimensional tilings), A060596A060602.
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


STATUS

approved



