The shape of permutominoes tiling the plane

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 XYX̂, where Xî 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 XYX̂ is pseudo-square is prefix of a unique infinite word Y with period 4|X|N|X|E. Also, we show that XYX̂ are centrosymmetric, i.e. they are fixed by rotation of angle π. The proof of this fact is based on the concept of pseudoperiods, a natural generalization of periods. © 2012 Elsevier B.V. All rights reserved.