Combinatorial aspects of L-convex polyominoes

We consider the class of L-convex polyominoes, i.e. those polyominoes in which any two cells can be connected with an “L” shaped path in one of its four cyclic orientations. The paper proves bijectively that the number fn of L-convex polyominoes with perimeter 2(n + 2) satisfies the linear recurrence relation fn+2 = 4 fn+1 − 2 fn , by first establishing a recurrence of the same form for the cardinality of the “2-compositions” of a natural number n, a simple generalization of the ordinary compositions of n. Then, such 2-compositions are studied and bijectively related to certain words of a regular language over four letters which is in turn bijectively related to L-convex polyominoes. In the last section we give a solution to the open problem of determining the generating function of the area of L-convex polyominoes. c © 2006 Elsevier Ltd. All rights reserved. 1. L-convex polyominoes: Basic definitions A polyomino is a finite union of elementary cells of the lattice Z × Z, whose interior is connected (see Fig. 1(a)). Polyominoes are defined up to translations. Invented by Golomb [14] who coined the term polyomino, these well-known combinatorial objects are related to many challenging problems, such as tilings [3,13], games [12], among many others. A column (row) of a polyomino is the intersection between the polyomino and an infinite strip of cells whose centers lie on a vertical (horizontal) line. In a polyomino the semi-perimeter is half the length of the border, while the area is the number of its cells. Enumerating polyominoes is one of the most famous problems in combinatorics, and despite several efforts made recently by physicists and mathematicians, it remains unsolved. The number E-mail addresses: [email protected] (G. Castiglione), [email protected] (A. Frosini), [email protected] (E. Munarini), [email protected] (A. Restivo), [email protected] (S. Rinaldi). 0195-6698/$ see front matter c © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ejc.2006.06.020 G. Castiglione et al. / European Journal of Combinatorics 28 (2007) 1724–1741 1725 Fig. 1. (a) A polyomino; (b) a convex polyomino; (c) an L-convex polyomino; (d) a stack polyomino. Fig. 2. (a) A path between two cells in a polyomino; (b) a monotone path made only of north and east steps, having four changes of direction. an of polyominoes with n cells is known up to n = 56 [17] and the asymptotic behavior of the sequence (an)n≥0 is partially known due to the relation limn→∞(an) = μ, where 3.96 < μ < 4.64, where the lower bound is a recent improvement of [2]. Nevertheless, several subclasses were enumerated by imposing convexity constraints [4,11]. A polyomino is h-convex (resp. v-convex) if each of its rows (resp. columns) is connected. A polyomino is hv-convex, or simply convex, if it is both h-convex and v-convex (see Fig. 1(b)). Let us now introduce the basic concept of our paper, i.e. the notion of k-convexity. We define a path in a polyomino to be a self-avoiding sequence of unit steps of four types: north N = (0, 1), south S = (0,−1), east E = (1, 0), and west W = (−1, 0), entirely contained in the polyomino. A path connecting two distinct cells, A and B , starts from the center of A, and ends at the center of B (see Fig. 2(a)). We say that a path is monotone if it is constituted only of steps of two types (see Fig. 2(b)). Given a path w = u1 . . . uk , each pair of steps ui ui+1 such that ui = ui+1, 0 < i < k, is called a change of direction. In [8] the authors observed that convex polyominoes have the property that every pair of cells is connected by a monotone path, and proposed a classification of convex polyominoes based on the number of changes of direction in the paths connecting any two cells of the polyomino. More precisely, a polyomino is k-convex if every pair of its cells can be connected by a monotone path with at most k changes of direction. For k = 1 we have the L-convex polyominoes, such that each two cells can be connected by a path with at most one change of direction (see Fig. 1(c)). In the sequel we present a different characterization of this class based on the notion of maximal rectangles [8]. A rectangle, that we denote by [x, y], with x, y ≥ 1, is a rectangular polyomino with x columns and y rows. For any polyomino P , we say that [x, y] is maximal in P if ∀[x ′, y ′], [x, y] ⊆ [x ′, y ′] ⊆ P ⇒ [x, y] = [x ′, y ′]. 1726 G. Castiglione et al. / European Journal of Combinatorics 28 (2007) 1724–1741 Fig. 3. (a), (b) Two rectangles having a crossing intersection; (c) two rectangles having a non-crossing intersection. Fig. 4. The L-convex polyomino in (a) can be defined by the overlapping of five non-comparable rectangles, each pair having crossing intersections, as depicted in (b). By abuse of notation, for any two polyominoes P and P ′ we will write P ⊆ P ′ to mean that P is geometrically included in P ′. Two rectangles [x, y] and [x ′y ′] contained in P have a crossing intersection if their intersection is a non-trivial rectangle whose basis is the smallest of the two bases and whose height is the smallest of the two heights, i.e. [x, y] ∩ [x ′, y ′] = [min(x, x ′),min(y, y ′)] ; see Fig. 3 for an example. The following theorem gives a characterization of L-convex polyominoes in terms of maximal rectangles [8]. Theorem 1. A convex polyomino P is L-convex if and only if any two of its maximal rectangles have a non-void crossing intersection. Thus, an L-convex polyomino can be seen as the overlapping of maximal rectangles (see Fig. 4 for an example). In the literature L-convex polyominoes have been considered from several points of view: in [9] it is shown that they are a well-ordering according to the sub-picture order; in [7] the authors have investigated some tomographical aspects, and have discovered that L-convex polyominoes are uniquely determined by their horizontal and vertical projections. Finally, in [6] it is proved that the number fn of L-convex polyominoes having semi-perimeter equal to n + 2 satisfies the recurrence relation fn+2 = 4 fn+1 − 2 fn (n ≥ 1), (1) with f0 = 1, f1 = 2, f2 = 7, giving the sequence 1, 2, 7, 24, 82, 280, 956, 3264, . . . (sequence A003480 in [22]). G. Castiglione et al. / European Journal of Combinatorics 28 (2007) 1724–1741 1727 A special class of L-convex polyominoes is the well-known class of stack polyominoes [19] [23, p. 76] [24], i.e. convex polyominoes such that the lowest row touches both the left and the right sides of the minimal bounding rectangle (see Fig. 1(d)). Indeed such polyominoes are characterized, among L-convex polyominoes, by the property that any two cells are connected by a path Eh1 Nh2 or Sh1 Eh2 , h1, h2 ≥ 0. That is, stack polyominoes are defined by two particular orientations of the L. In this paper we extend the study of the combinatorial properties of the class of L-convex polyominoes and of the sequence ( fn)n≥0. In particular, the main results of the paper are the following: 1. in Section 2 we introduce and study the class of 2-compositions, a natural extension of the ordinary compositions; in particular we prove that this class is enumerated by the sequence ( fn)n≥0, and then we establish several properties of this sequence; 2. in Section 3 we determine a bijection between 2-compositions and L-convex polyominoes, thus giving a combinatorial explanation that L-convex polyominoes satisfy the simple recurrence in (1); such a bijection allows us also to count L-convex polyominoes according to other interesting statistics, such as the number of rows and columns and the number of maximal rectangles; 3. in Section 4, we determine the generating function for L-convex polyominoes according to the area; this problem was also considered in [6] using the ECO method, which is a constructive method for producing all the objects of a given class, according to the growth of a certain parameter (the size) of the objects; basically, the idea is to perform local expansions on each object of size n, thus constructing a set of objects of the successive size (see [1] for more details); the authors of [6] determined a system of algebraic equations involving the area generating function, without solving it by means of other than standard analytical techniques. Finally we point out that some unpublished results on the class of L-convex polyominoes have been obtained independently by W. James in [16]. In particular W. James introduces a general method of representing polyominoes by their column structure (the so called dot notation), and applies this method to determine the generating function for the area of L-convex polyominoes and to discover some combinatorial properties of the class. 2. The class of 2-compositions A composition of a natural number n is an ordered partition of n, that is a k-tuple (x1, . . . , xk) of positive integers such that x1 + · · · + xk = n (see [10]). We now extend the definition of composition to the 2-dimensional case. A 2-composition of n is a 2 × k matrix whose entries are non-negative integers that add up to n and such that each column has at least one non-zero entry. The number of columns, k ≥ 0, is called the length of the 2-composition. Let Un be the class of 2-compositions of n and let un = |Un|. Notice that U0 contains only the empty 2-composition, while U1 = {[ 0 1 ] , [ 1 0 ]} , U2 = {[ 0 2 ] , [ 1 1 ] , [ 2 0 ] , [ 1 0 0 1 ] , [ 0 1 1 0 ] , [ 1 1 0 0 ] , [ 0 0 1 1 ]} . Hence u0 = 1, u1 = 2 and u2 = 7. More generally we have the following result. Proposition 1. The numbers un satisfy the recurrence un+2 = 4un+1 − 2un (2) for n ≥ 1, with the initial values u0 = 1, u1 = 2, u2 = 7. 1728 G. Castiglione et al. / European Journal of Combinatorics 28 (2007) 1724–1741 Proof. Let n ≥ 1. The 2-compositions in Un+2 can be all obtained by performing the following operations on each 2-composition M ∈ Un+1: 1. add a column [ 1 0 ] on the left of M; 2. add a column [ 0 1 ] on the left of M; 3. increase by one the first element on the first row of M; 4. increase by one the first element on the second row of M . By performing the four operations on the 2-compositions of Un+1 we obtain a set of 4un+1 elements of Un+2. However, some 2-compositions are obtained twice, and they are precisely those containing no null elements in the first column, that is: 1. those whose first column is [ 1 1 ] ; 2. those whose first column is [ x + 1 y + 1 ] , with x, y ≥ 0 and (x, y) = (0, 0). Since the number of elements in these two classes is clearly given by 2un it follows that un+2 = 4un+1 − 2un . Thus we have the remarkable fact that the number of the L-convex polyominoes with semiperimeter n+2 is equal to the number of the 2-compositions of n. In Section 3 we will determine a simple bijection between these two classes. As proved in the following proposition, the numbers un also satisfy a recurrence which is not linear yet has positive constant coefficients. Proposition 2. The numbers un satisfy the recurrence un+2 = 3un+1 + un + un−1 + · · · + u0 (3) with the initial values u0 = 1 and u1 = 2. Proof. The 2-compositions of n + 2 can be partitioned according to the form of the first column. Indeed the first column can be: 1. [ 0 y ] with y = 1, 2, . . . , n + 2; deleting from each the first column, we obtain Un+2−k ; 2. [ 1 0 ] ; deleting from each the first column, we obtain Un+1; 3. [ x + 2 0 ] with x ≥ 0 or [ y z ] with y, z ≥ 1; subtracting 1 from the top entry of the first column, we obtain Un+1. This implies recurrence (3). Moreover we have that the numbers un satisfy a recurrence relation with non-constant coefficients. Proposition 3. The numbers un satisfy the recurrence un+1 = n ∑ k=0 (k + 2)un−k (4) with the initial value u0 = 1. G. Castiglione et al. / European Journal of Combinatorics 28 (2007) 1724–1741 1729 Fig. 5. On the left: table of the numbers un,k . On the right: table of the numbers vn,k , n, k ≥ 6. Proof. Consider those 2-compositions of n + 1 for which the first column is [ x k − x + 1 ] with fixed values of k and x and 0 ≤ k ≤ n, 0 ≤ x ≤ k + 1. Deleting these first columns we obtain Un−k . This happens k + 2 times (for each x = 0, 1, . . . , k + 1). Hence identity (4). The numbers un satisfy the following property which resembles the well-known Cassini identity for Fibonacci numbers [18]. Proposition 4. The numbers un have the property un+1 − un+2un = 2n−1 (n ≥ 1). (5)