Tiling the plane with one tile
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
On the tiling by translation problem
Discrete Applied Mathematics
Tiling the plane with permutations
DGCI'11 Proceedings of the 16th IAPR international conference on Discrete geometry for computer imagery
A parallelogram tile fills the plane by translation in at most two distinct ways
Discrete Applied Mathematics
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In this paper we explore the connections between two classes of polyominoes, namely the permutominoes and the pseudo-square polyominoes. A permutomino is a polyomino uniquely determined by a pair of permutations. Permutominoes, and in particular convex permutominoes, have been considered in various kinds of problems such as: enumeration, tomographical reconstruction, and algebraic characterization. On the other hand, pseudo-square polyominoes are a class of polyominoes tiling the plane by translation. The characterization of such objects has been given by Beauquier and Nivat, who proved that a polyomino tiles the plane by translation if and only if it is a pseudo-square or a pseudo-hexagon. In particular, a polyomino is pseudo-square if its boundary word may be factorized as XYX@^Y@^, where X@^ denotes the path X traveled in the opposite direction. In this paper we relate the two concepts by considering the pseudo-square polyominoes which are also convex permutominoes. By using the Beauquier-Nivat characterization we provide some geometrical and combinatorial properties of such objects, and we show for any fixed X, each word Y such that XYX@^Y@^ is pseudo-square is prefix of a unique infinite word Y"~ with period 4|X|"N|X|"E. Also, we show that XYX@^Y@^ are centrosymmetric, i.e. they are fixed by rotation of angle @p. The proof of this fact is based on the concept of pseudoperiods, a natural generalization of periods.