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{\bf Srecko Brlek, Andrea Frosini, Simone Rinaldi and Laurent Vuillon}
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{\bf Tilings by Translation: Enumeration by a Rational Language Approach}
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Beauquier and Nivat introduced and gave a characterization of the
class of {\em 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 {\em PSP} of pseudo-square polyominoes which are also {\em
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 {\em convex} polyominoes
is also analyzed.
\bye