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%I #39 Aug 06 2023 08:18:42
%S 1,2,20,7658,12954016496,10592917773063552232751878
%N Number of tilings of the 8-dimensional zonotope constructed from D vectors.
%C 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.
%C 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
%D 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.
%D 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.
%H N. Destainville, R. Mosseri and F. Bailly, <a href="https://arxiv.org/abs/cond-mat/0004145">Fixed-boundary octagonal random tilings: a combinatorial approach</a>, arXiv:cond-mat/0004145 [cond-mat.stat-mech], 2000.
%H N. Destainville, R. Mosseri and F. Bailly, <a href="https://doi.org/10.1023/A:1026564710037">Fixed-boundary octagonal random tilings: a combinatorial approach</a>, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
%H S. Felsner and H. Weil, <a href="http://doi.org/10.1016/S0166-218X(00)00232-8">Sweeps, arrangements and signotopes</a>, Discrete Applied Mathematics, Volume 109, Issues 1-2, 2001, Pages 67-94.
%H M. Latapy, <a href="https://arxiv.org/abs/math/0008022">Generalized Integer Partitions, Tilings of Zonotopes and Lattices</a>, arXiv:math/0008022 [math.CO], 2000.
%H G. M. Ziegler, <a href="https://www.mi.fu-berlin.de/math/groups/discgeom/ziegler/Preprintfiles/025PREPRINT.pdf">Higher Bruhat Orders and Cyclic Hyperplane Arrangements</a>, Topology, Volume 32, 1993.
%F 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
%e 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.
%Y Cf. A006245 (two-dimensional tilings), A060595-A060602.
%Y Column k=8 of A060637.
%K nonn,nice,hard,more
%O 8,2
%A Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 12 2001
%E a(12) from _Manfred Scheucher_, Nov 30 2021
%E Edited by _Manfred Scheucher_, Mar 08 2022
%E a(13) from _Manfred Scheucher_, Aug 06 2023
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