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A060602 Number of tilings of the d-dimensional zonotope constructed from d+3 vectors. 10
8, 24, 62, 148, 338, 752, 1646, 3564, 7658, 16360, 34790, 73700, 155618, 327648, 688094, 1441756, 3014618, 6291416, 13107158, 27262932, 56623058, 117440464, 243269582, 503316428, 1040187338, 2147483592, 4429184966 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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 the codimension, i.e., D-d, is constant = 3 and d >= 0.
Also the number of signotopes on r+2 elements of rank r. A signotope on n elements 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
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
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.
FORMULA
Conjectures from Colin Barker, Feb 20 2013: (Start)
a(n) = 2*(-3+7*2^n+(-1+2^n)*n).
G.f.: -2*(4*x^3-11*x^2+12*x-4) / ((x-1)^2*(2*x-1)^2). (End)
The above conjectures are correct; see Proposition 7.1 in Ziegler's article. - Manfred Scheucher, Feb 09 2022
a(n) = 2 * A133546(n+2). - Alois P. Heinz, Feb 11 2022
EXAMPLE
For any Z(D,d), the number of codimension 0 tilings is always 1, with codimension 1 it is 2, with codimension 2 it is 2.D.
MATHEMATICA
LinearRecurrence[{6, -13, 12, -4}, {8, 24, 62, 148}, 30] (* Harvey P. Dale, Oct 13 2023 *)
PROG
(Python) print([2**(n + 1)*(n + 7) - 2*n - 6 for n in range(100)])
CROSSREFS
Cf. A006245 (two-dimensional tilings), A060595-A060601. A diagonal of A060637. See also A351383 and A351384 for other diagonals.
Cf. A133546.
Sequence in context: A177719 A317234 A049724 * A066605 A066497 A205963
KEYWORD
nonn,nice,easy
AUTHOR
Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 12 2001
EXTENSIONS
Edited by Manfred Scheucher, Mar 08 2022
STATUS
approved

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Last modified February 28 15:18 EST 2024. Contains 370400 sequences. (Running on oeis4.)