A060601 Number of tilings of the 9-dimensional zonotope constructed from D vectors.

1, 2, 22, 16360, 613773463394

Offset: 9

Author

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

Keywords

• nonn
• nice

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=9 and D varies.

Also the number of signotopes of rank 10. 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

Examples

			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.
		

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, 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.
• G. M. Ziegler, Higher Bruhat Orders and Cyclic Hyperplane Arrangements, Topology, Volume 32, 1993.

Crossrefs

Cf. A006245 (two-dimensional tilings), A060595-A060602.

Column k=9 of A060637.

Formula

Asymptotics: a(n) = 2^(Theta(n^9)). 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^9} <= a(n) <= 2^{d n^9} is satisfied. - Manfred Scheucher, Sep 22 2021

Extensions

a(13) from Manfred Scheucher, Mar 07 2022

Edited by Manfred Scheucher, Mar 08 2022