The number of directed k-convex polyominoes

In the plane Z× Z a cell is a unit square and a polyomino is a finite connected union of cells. Polyominoes are defined up to translations. Since they have been introduced by Golomb [20], polyominoes have become quite popular combinatorial objects and have shown relations with many mathematical problems, such as tilings [6], or games [19] among many others. Two of the most relevant combinatorial problems concern the enumeration of polyominoes according to their area (i.e., number of cells) or semi-perimeter. These two problems are both difficult to solve and still open. As a matter of fact, the number an of polyominoes with n cells is known up to n = 56 [21] and asymptotically, these numbers satisfy the relation limn(an) 1/n = μ, 3.96 < μ < 4.64, where the lower bound is a recent improvement of [4]. In order to probe further, several subclasses of polyominoes have been introduced on which to hone enumeration techniques. Some of these subclasses can be defined using the notions of connectivity and directedness: among them we recall the convex, directed, parallelogram polyominoes, which will be considered in this paper. Formal definitions of these classes will be given in the next section. In the literature, these objects have been widely studied using different techniques. Here, we just outline some results which will be useful for the reader of this paper: