2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 145150
author:  Éric Rémila 

title:  Structure of spaces of rhombus tilings in the lexicograhic case 
keywords:  rhombus tiling, flip, connectivity 
abstract: 
Rhombus tilings are tilings of zonotopes with rhombohedra.
We study a class of lexicographic rhombus tilings
of zonotopes, which are deduced from higher Bruhat orders
relaxing the unitarity condition. Precisely, we fix a
sequence
(v
of vectors of
1
, v
2
,…, v
D
)
ℝ
and a sequence
d
(m
of positive integers. We assume (lexicographic
hypothesis) that for each subsequence
1
, m
2
,…, m
D
)
(v
of length
i
, v
1
i
,…, v
2
i
)
d
d
, we have
det(v
. The zonotope
i
, v
1
i
,…, v
2
i
) > 0
d
Z
is the set
{ Σα
. Each prototile used in a tiling of
i
v
i
0 ≤α
i
≤m
i
}
Z
is a rhombohedron constructed from a subsequence of
d
vectors. We prove that the space of tilings of
Z
is a graded poset, with minimal and maximal element.

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reference:  Éric Rémila (2005), Structure of spaces of rhombus tilings in the lexicograhic case, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 145150 
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