Daniela Battaglino ; Jean-Marc Fédou ; Simone Rinaldi ; Samanta Socci - The number of $k$-parallelogram polyominoes

dmtcs:2370 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.2370
The number of $k$-parallelogram polyominoesArticle

Authors: Daniela Battaglino 1,2; Jean-Marc Fédou 1; Simone Rinaldi ORCID2; Samanta Socci 2

  • 1 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe MC3
  • 2 Dipartimento di Matematica e Informatica [Siena]

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 $\textit{Fibonacci polynomials}$.


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: Convex polyominoes,L-convex polyominoes,Rational generating functions,Fibonacci polynomials,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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