cp's OEIS Frontend

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

%I A060595 #50 Jul 24 2023 15:09:50
%S A060595 1,2,10,148,7686,1681104,1881850464,13227777493060
%N A060595 Number of tilings of the 3-dimensional zonotope constructed from D vectors.
%C A060595 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.
%C A060595 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
%D A060595 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 A060595 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.
%H A060595 Helena Bergold, Stefan Felsner, and Manfred Scheucher, <a href="http://page.math.tu-berlin.de/~scheuch/publ/bfs-ehds-eurocg22.pdf">Extendability of higher dimensional signotopes</a>, Proc. 38th Eur. Wksp. Comp. Geom. (EuroCG), 2022. See also <a href="https://arxiv.org/abs/2303.04079">arXiv:2303.04079</a> [math.CO], 2023.
%H A060595 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 A060595 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 A060595 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 A060595 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 A060595 J. A. Olarte and F. Santos, <a href="https://arxiv.org/abs/1906.05764">Hypersimplicial subdivisions</a>, arXiv:1906.05764 [math.CO], 2019.
%H A060595 Manfred Scheucher, <a href="/A060595/a060595_1.cpp.txt">C program for enumeration</a>
%H A060595 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 A060595 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
%e A060595 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.
%Y A060595 Cf. A006245 (two-dimensional tilings), A060596-A060602.
%Y A060595 Column k=3 of A060637.
%K A060595 nonn,nice
%O A060595 3,2
%A A060595 Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 12 2001
%E A060595 a(8)-a(9) from _Manfred Scheucher_, Sep 13 2021
%E A060595 Edited by _Manfred Scheucher_, Mar 08 2022
%E A060595 a(10) from _Manfred Scheucher_, Jul 17 2023