The number of directed -convex polyominoes
In the plane 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 , polyominoes have become quite popular combinatorial objects and have shown relations with many mathematical problems, such as tilings , or games  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 of polyominoes with cells is known up to  and asymptotically, these numbers satisfy the relation , where the lower bound is a recent improvement of .
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:
In this paper we present a new general approach for the enumeration of directed convex polyominoes, which let us easily control several statistics. This method relies on a bijection between directed convex polyominoes with semi-perimeter equal to and triplets , where and are forests of and trees, respectively, with a total number of nodes equal to , and is a lattice path made of east and south unit steps (see Proposition 3).
Basing 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 considered in the literature (see, for instance [3, 7, 22]), but what makes our method interesting, to our opinion, is that every statistic which can be read on the two forests and , can be in principle computed. Basically, with being a class of directed-convex polyominoes, our method consists of three steps:
provide a characterization of the two forests and of the path , for the polyominoes in ;
determine the generating function – according to the considered statistics – of the forests and of the paths , for the polyominoes in ;
obtain the generating function of by means of the composition of the generating function of the paths with the generating functions of the trees of .
Furthermore, the previously described bijection can be easily translated into a (new) bijection between directed convex polyominoes and Grand-Dyck paths (or bilateral Dyck paths). Other bijections between these two classes of objects can be found in the literature, see for instance,  and .
The most important and original result of the paper consists in applying our method to the enumeration of directed -convex polyominoes, i.e. directed convex polyominoes which are also -convex. Let us recall that in  it was proposed a classification of convex polyominoes based on the number of changes of direction in the paths connecting any two cells of a polyomino. More precisely, a convex polyomino is -convex if every pair of its cells can be connected by a monotone path with at most changes of direction.
For we have the -convex polyominoes, where any two cells can be connected by a path with at most one change of direction. In the recent literature -convex polyominoes have been considered from several points of view: in  it is shown that they are a well-ordering according to the sub-picture order; in  the authors have investigated some tomographical aspects, and have discovered that -convex polyominoes are uniquely determined by their horizontal and vertical projections. Finally, in [12, 13] it is proved that the number of -convex polyominoes having semi-perimeter equal to satisfies the recurrence relation
For we have -convex (or -convex) polyominoes, such that each two cells can be connected by a path with at most two changes of direction. Unfortunately, -convex polyominoes do not inherit most of the combinatorial properties of -convex polyominoes. In particular, their enumeration resisted standard enumeration techniques and it was obtained in  by applying the so-called inflation method. The authors proved that the generating function of -convex polyominoes with respect to the semi-perimeter is:
where is the generating function of Catalan numbers . Hence the number of Z-convex polyominoes having semi-perimeter grows asymptotically as . However, the solution found for -convex polyominoes seems to be not easily generalizable to a generic , hence the problem of enumerating -convex polyominoes for is still open and difficult to solve. Recently, some efforts in the study of the asymptotic behavior of -convex polyominoes have been made by Micheli and Rossin in .
Thus, in order to simplify the problem of enumerating polyominoes satisfying the -convexity constraint, the authors of  provide a general method to enumerate an important subclass of -convex polyominoes, that of -parallelogram polyominoes, i.e. -convex polyominoes which are also parallelogram. In this case, the authors prove that the generating function is rational for any , and it can be expressed as follows:
where are the Fibonacci polynomials are defined by the following recurrence relation
At the end of  the authors observe that in  it is shown that the generating function of trees with height less than or equal to can be also expressed in terms of the Fibonacci polynomials, and it is precisely equal to
Thus, considering (2), it was proposed the problem of finding a bijection between -parallelogram polyominoes and pairs of trees having height at most , minus pairs of trees having both height exactly equal to .
As already mentioned, in this paper we succeed in solving this problem, as a specific sub-case of a more general framework: indeed we are able to provide a bijective proof for the number of directed -convex polyominoes. In order to reach this goal, we rely on the observation that our bijection can be used to express the convexity degree of the polyomino in terms of the heights of the trees of the forests and . Basing on this observation, the application of our method eventually let us determine the generating function of directed -convex polyominoes, for any . This is a rational function, which be suitably expressed, again, in terms of the Fibonacci polynomials. More precisely, we prove that, for every , the generating function of directed -convex polyominoes according to the semi-perimeter is equal to:
We point out that (3) is the first result in the literature concerning the -convexity constraint on directed-convex polyominoes. Due to the rationality of the generating function we can then apply standard techniques to provide the asymptotic behavior for the number of directed -convex polyominoes.
Moreover, if we restrict ourselves to the case of parallelogram polyominoes, we observe that our bijection reduces to a bijection between parallelogram polyominoes with semi-perimeter and pairs of trees, and this gives a simple tool to present a combinatorial proof for (1).
2 Notation and preliminaries
In this section we recall some basic definitions about polyominoes, and other combinatorial objects, which will be used in the rest of the paper. First, we point out that, in the representaion of polyominoes, we use the convention that the south-west corner of the minimal bounding rectangle is placed at . A column (row) of a polyomino is the intersection of the polyomino and an infinite strip of cells whose centers lie on a vertical (horizontal) line. A polyomino is said to be column-convex (row-convex) when its intersection with any vertical (horizontal) line is connected. Figure 1, show a column convex and a row convex polyomino, respectively. A polyomino is convex if it is both column and row convex (see Fig. 1). The semi-perimeter of a polyomino is naturally defined as half the perimeter, while the area is the number of its cells. As a matter of fact, the semi-perimeter of a convex polyomino is given by the sum of the number of its rows and its columns.
Let be a convex polyomino whose minimal bounding rectangle has dimension . We number the columns and the rows from left to right and from bottom to top, respectively. Thus, we consider the bottom (resp. top) row of as its first (resp. last) row, and the leftmost (resp. rightmost) column of as its first (resp. last) column. By convention, we will often write to denote the cell of , whose north-west corner has coordinates equal to . With that convention, the cell is at the intersection of the -th column and the -th row.
A path is a self-avoiding sequence of unit steps of four types: north , south , east , and west . A path is said to be monotone if it is comprised only of steps of two types.
A polyomino is said to be directed when each of its cells can be reached from a distinguished cell, called the root (denoted by ), by a path which is contained in the polyomino and uses only north and east unit steps. A polyomino is directed convex if it is both directed and convex.
We recall that a parallelogram polyomino is a polyomino whose boundary can be decomposed in two paths, the upper and the lower paths, which are comprised of north and east unit steps and meet only at their starting and final points. It is clear that parallelogram polyominoes are also directed convex polyominoes, while the converse does not hold.
Figure 2 shows all directed convex polyominoes with semi-perimeter equal to , where only the rightmost polyomino is not a parallelogram one.
Now, we introduce another class of objects, which will be useful in this paper. An ordered tree is a rooted tree for which an ordering is specified for the children of each vertex. In this paper we shall say simply tree instead of ordered tree. The size of a tree is the number of nodes. The height of a tree is the number of nodes on a maximal simple path starting at the root of .