Tilings by Translation: Enumeration by a Rational Language Approach

Beauquier and Nivat introduced and gave a characterization of the class of pseudo-square polyominoes, i.e. those polyominoes that tile the plane by translation: a polyomino tiles the plane by translation if and only if its boundary word $W$ may be factorized as $W = XY\overline{X} \,\overline{Y}$. In this paper we consider the subclass PSP of pseudo-square polyominoes which are also parallelogram. By using the Beauquier-Nivat characterization we provide by means of a rational language the enumeration of the subclass of $psp$-polyominoes with a fixed planar basis according to the semi-perimeter. The case of pseudo-square convex polyominoes is also analyzed.