The number of $k$-parallelogram polyominoes

Abstract : A convex polyomino is $k$-$\textit{convex}$ if every pair of its cells can be connected by means of a $\textit{monotone path}$, internal to the polyomino, and having at most $k$ changes of direction. The number $k$-convex polyominoes of given semi-perimeter has been determined only for small values of $k$, precisely $k=1,2$. In this paper we consider the problem of enumerating a subclass of $k$-convex polyominoes, precisely the $k$-$\textit{convex parallelogram polyominoes}$ (briefly, $k$-$\textit{parallelogram polyominoes}$). For each $k \geq 1$, we give a recursive decomposition for the class of $k$-parallelogram polyominoes, and then use it to obtain the generating function of the class, which turns out to be a rational function. We are then able to express such a generating function in terms of the $\textit{Fibonacci polynomials}$.
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Daniela Battaglino, Jean-Marc Fédou, Simone Rinaldi, Samanta Socci. The number of $k$-parallelogram polyominoes. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.1113-1124. ⟨hal-01229685⟩

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