

A060598


Number of tilings of the 7dimensional zonotope constructed from D vectors.


1




OFFSET

7,2


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=7 and D varies.
Also the number of signotopes of rank 8. 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(Ii_0), X(Ii_1), ..., X(Ii_r) changes its sign at most once (see FelsnerWeil 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, 19961997), 293336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.


LINKS

Table of n, a(n) for n=7..11.
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.
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^7)). 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^7} <= a(n) <= 2^{d n^7} 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 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), A060595A060602.
Column k=7 of A060637.
Sequence in context: A076954 A206847 A259654 * A055687 A006262 A003043
Adjacent sequences: A060595 A060596 A060597 * A060599 A060600 A060601


KEYWORD

nonn,nice


AUTHOR

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


EXTENSIONS

a(11) from Manfred Scheucher, Sep 22 2021
Edited by Manfred Scheucher, Mar 08 2022


STATUS

approved



