Abstract
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.
Highlights
In the plane Z × Z a cell is a unit square and a polyomino is a finite connected union of cells
In this paper we present a new bijection between directed convex polyominoes and triplets (Fe, Fs, λ), where Fe and Fs are forests of trees, and λ is a lattice path made of two types of steps, satisfying special constraints
Lemma 2 The degree of convexity of a parallelogram polyomino is equal to the number of changes of direction of the minimal bounce path m of the parallelogram polyomino
Summary
In the plane Z × Z a cell is a unit square and a polyomino is a finite connected union of cells (see Fig. 1). In this paper we present a new bijection between directed convex polyominoes and triplets (Fe, Fs, λ), where Fe and Fs are forests of trees, and λ is a lattice path made of two types of steps, satisfying special constraints This bijection allows to express the convexity degree of the polyomino in terms of the heights of the trees of Fe and Fs. This bijection allows to express the convexity degree of the polyomino in terms of the heights of the trees of Fe and Fs Basing of this bijection, we develop a new method, for the enumeration of directed convex polyominoes, which allows us to control several statistics, including the semi-perimeter, the degree of convexity, the width, the height, the size of the last row/column and the number of corners. We point out that the full version of the paper containing all the definitions and proofs, with several examples is on the ArXiv [7]
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