n this paper we consider the class of column-convex permutominoes, i.e. column-convex polyominoes defined by a pair of permutations (π 1 , π 2 ). First, using a geometric construction, we prove that for every permutation π there is at least one column-convex permutomino P such that π 1 (P ) = π or π 2 (P ) = π. In the second part of the paper, we show how, for any given permutation π, it is possible to define a set of logical implications F(π) on the points of π, and prove that there exists a column-convex permutomino P such that π 1 (P ) = π if and only if F(π) is satisfiable. This property can be then used to give a characterization of the set of column-convex permutominoes P such that π 1 (P ) = π.
International audience 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 $\textit{Fibonacci polynomials}$. Un polyomino convexe est dit $k$-$\textit{convexe}$ lorsqu’on peut relier tout couple de cellules par un chemin monotone ayant au plus $k$ changements de direction. Le nombre de polyominos $k$-convexes n’est connu que pour les petites valeurs de $k = 1,2$. Dans cet article, nous énumérons la sous-classe des polyominos $k$-convexes qui sont également parallélogramme, que nous appelons $k$-$\textit{parallélogrammes}$. Nous donnons une décomposition récursive de la classe des polyominos $k$-parallélogrammes pour chaque $k$, et en déduisons la fonction génératrice, rationnelle, selon le demi-périmètre. Nous donnons enfin une expression de cette fonction génératrice en termes des $\textit{polynômes de Fibonacci}$.
The notion of a pattern within a binary picture (polyomino) has been introduced and studied in [3], and resembles the notion of pattern containment within permutations. The main goal of this paper is to extend the studies of [3] by adopting a more geometrical approach: we use the notion of pattern avoidance in order to recognize or describe families of polyominoes defined by means of geometrical constraints or combinatorial properties. Moreover, we extend the notion of pattern in a polyomino, by introducing generalized polyomino patterns, so that to be able to describe more families of polyominoes known in the literature.
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International audience We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed $k$-convex polyominoes.We show it is a rational function and we study its asymptotic behavior. Nous présentons une nouvelle méthode générique pour obtenir facilement et rapidement les fonctions génératrices des polyominos dirigés convexes avec différentes combinaisons de statistiques : hauteur, largeur, longueur de la dernière ligne/colonne et nombre de coins. La méthode peut être utilisée pour énumérer différentes familles de polyominos dirigés convexes: les polyominos symétriques, les polyominos parallélogrammes. De cette façon, nouscalculons la fonction génératrice des polyominos dirigés $k$-convexes, nous montrons qu’elle est rationnelle et nous étudions son comportement asymptotique.
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