The number of directed [formula omitted]-convex polyominoes

Abstract

We present a new general approach for the enumeration of directed convex polyominoes, which lets us easily control several statistics. This method relies on a bijection between directed convex polyominoes with semi-perimeter equal to n+2 and triplets (Fe,Fs,λ), where Fe and Fs are forests with a total number of nodes equal to n, and λ is a lattice path of length n+2. Based on this bijection, we develop a new method for the enumeration of directed convex polyominoes, according to several different parameters, 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 most of these statistics have already been computed in the literature, but applying our method every statistic which can be read on the two forests Fe and Fs, can be in principle computed. The most important and original result of the paper consists of applying our method to the enumeration of directed convex polyominoes which are also k-convex. Let us recall that a convex polyomino is k-convex if every pair of its cells can be connected by a monotone path with at most k changes of direction. The application of our method eventually lets us prove that the generating function of directed k-convex polyominoes, for any k≥1, is a rational function, which can be suitably expressed in terms of the known Fibonacci polynomials Fk: z2Fk+2F2k+32F2k+2.