A060602 Number of tilings of the d-dimensional zonotope constructed from d+3 vectors.

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

Offset: 0

Author

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

Keywords

• nonn
• nice
• easy

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

Examples

			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.
		

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.
• Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Crossrefs

Cf. A006245 (two-dimensional tilings), A060595-A060601. A diagonal of A060637. See also A351383 and A351384 for other diagonals.

Cf. A133546.

Programs

Mathematica

LinearRecurrence[{6,-13,12,-4},{8,24,62,148},30] (* Harvey P. Dale, Oct 13 2023 *)

Python

print([2**(n + 1)*(n + 7) - 2*n - 6 for n in range(100)])

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

Extensions

Edited by Manfred Scheucher, Mar 08 2022