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A060637 Triangle T(n,k) (0 <= k <= n) giving number of tilings of the k-dimensional zonotope constructed from n vectors.

%I A060637 #22 Mar 09 2022 01:47:54
%S A060637 1,2,1,4,2,1,8,6,2,1,16,24,8,2,1,32,120,62,10,2,1,64,720,908,148,12,2,
%T A060637 1,128,5040,24698,7686,338,14,2,1,256,40320,1232944
%N A060637 Triangle T(n,k) (0 <= k <= n) giving number of tilings of the k-dimensional zonotope constructed from n vectors.
%C A060637 The zonotope Z(n,k) is the projection of the n-dimensional hypercube onto the k-dimensional space and the tiles are the projections of the k-dimensional faces of the hypercube.
%C A060637 T(n,k) is also the number of signotopes on n elements of rank r=k+1. 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
%D A060637 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 A060637 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 A060637 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 A060637 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 A060637 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 A060637 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.
%e A060637 Triangle T(n,k) begins:
%e A060637     1;
%e A060637     2,    1;
%e A060637     4,    2,     1;
%e A060637     8,    6,     2,    1;
%e A060637    16,   24,     8,    2,   1;
%e A060637    32,  120,    62,   10,   2,  1;
%e A060637    64,  720,   908,  148,  12,  2, 1;
%e A060637   128, 5040, 24698, 7686, 338, 14, 2, 1;
%e A060637   ...
%Y A060637 Diagonals give A000079, A000142, A006245, A060595-A060602, A351383, A351384. Cf. A060638.
%K A060637 nonn,tabl,hard,nice
%O A060637 0,2
%A A060637 _N. J. A. Sloane_, Apr 16 2001
%E A060637 Edited by _Manfred Scheucher_, Mar 08 2022