Tiling the Plane with Permutations
نویسندگان
چکیده
A permutomino is a polyomino uniquely defined by a pair of permutations. Recently permutominoes, and in particular convex permutominoes have been studied by several authors concerning their analytical and bijective enumeration, tomographical reconstruction, and the algebraic characterization of the associated permutations [2, 3]. On the other side, Beauquier and Nivat [5] introduced and gave a characterization of the class of pseudo-square polyominoes, i.e. polyominoes that tile the plane by translation: a polyomino is pseudo-square if its boundary word W may be factorized as W = XYX Y . In this paper we consider 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 that the boundary of a pseudo-square convex permutomino can be represented in terms of words X and Y which satisfy special constraints and which are periodic words with period 4 |X| |Y |. Several conjectures obtained through exhaustive search are also presented in the final section.
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تاریخ انتشار 2011