Counting k-Convex Polyominoes

We compute an asymptotic estimate of a lower bound of the number of k-convex polyominoes of semiperimeter p. This approximation can be written as μ(k)p4p where μ(k) is a rational fraction of k which up to μ(k) is the asymptotics of convex polyominoes. A polyomino is a connected set of unit square cells drawn in the plane Z × Z [7]. The size of a polyomino is the number of its cells. A central problem, which proved to be difficult, is to find exactly or even asymptotically A(n), the number a polyominoes of size n. Klarner proves that limn→∞ n √ A(n) exists and is upper bounded by 4.64 [8]. The lower bound was recently improved up to 3.98 [1]. In order to approximate the number of polyominoes, subclasses have been introduced, one of them being convex polyominoes. A horizontal, resp. vertical, convex polyomino is a polyomino for which each row, resp. column, is convex. Figure 1 gives an example of horizontal but not vertical convex polyomino and of a horizontal and vertical convex polyomino called convex polyomino. Bender gives an asymptotic estimate fr−n of the number of convex polyominoes with n cells, f and r being numerical constants [2]. Convex polyominoes are often considered according to number of rows and columns, called semiperimeter. Delest and Viennot [5] prove that the number of convex polyominoes of semiperimeter p+ 4 is fp+4 = (2p+ 11)4 p − 4(2p+ 1) ( 2p p ) ∼ 2p4. In [3], Castiglione and Restivo observe that in a convex polyomino each pair of cells can be joined by a monotone path (monotone means that it contains only two kinds of steps, e.g. East (1, 0) and North (0, 1) or East and South (0,−1)). The minimal number ∗This work was completed with the support of the ANR (project MAGNUM ANR-2010-BLAN-0204). the electronic journal of combinatorics 20(2) (2013), #P56 1 Figure 1: a horizontal (but not vertical) convex polyomino and a convex polyomino k of turns in the monotone paths linking two cells gives rise to a parameter which we call the complexity, and is the basis of how we shall classify polyominoes here. A polyomino is called k-convex if every pair of cells can be connected with a path of complexity at most k and there exists a path of complexity k linking two cells. Examples of k-convex (k = 0, 1, 2) polyominoes are given in Figure 2. 1-convex polyominoes are also called L-convex [3], 2-convex polyominoes are also denoted Z-convex. The generating function for L-convex polyominoes where the variable t marks semiperimeter is given by G(t) = 1−2t+t 2 1−4t+2t2 [4]. Z-convex polyominoes have been recently studied by Duchi, Rinaldi and Schaeffer [6] who compute the rational generating function and provide an asymptotics in p 24 4 when the semiperimeter is p+ 2. In this article we provide a lower bound for the number of k-convex polyominoes for any k and show that the asymptotics is also μ(k)p4. The key idea in this article is to transform a random walk into a set of polyominoes. Random walks are chosen with small deviation. Thus the boundary of the obtained polyominoes is delimited by two rectangles and we show that in that case, the polyominoes are k-convex. The first section gives basic definitions and first results on random walks with small deviations and polyominoes. In the second section, we describe the algorithm that tranform one random walk into a set of bounded polyominoes and prove its correctness. Finally, in the last section, we give the asymptotic number of a lower bound of k-convex polyominoes. 1 Small deviation random walks and polyominoes