ar X iv : 0 71 1 . 05 82 v 1 [ m at h . C O ] 5 N ov 2 00 7 PERMUTATIONS DEFINING CONVEX PERMUTOMINOES

A permutomino of size n is a polyomino determined by particular pairs (π1, π2) of permutations of size n, such that π1(i) 6= π2(i), for 1 ≤ i ≤ n. Here we determine the combinatorial properties and, in particular, the characterization for the permutations defining convex permutominoes. Using such a characterization, these permutations can be uniquely represented in terms of the so called square permutations, introduced by Mansour and Severini. Then, we provide a closed formula for the number of these permutations with size n. 1. Convex polyominoes In the plane Z×Z a cell is a unit square, and a polyomino is a finite connected union of cells having no cut point. Polyominoes are defined up to translations (see Figure 1 (a)). A column (row) of a polyomino is the intersection between the polyomino and an infinite strip of cells lying on a vertical (horizontal) line. Polyominoes were introduced by Golomb [17], and then they have been studied in several mathematical problems, such as tilings [2, 16], or games [15] among many others. The enumeration problem for general polyominoes is difficult to solve and still open. The number an of polyominoes with n cells is known up to n = 56 [18] 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 [1]. In order to simplify enumeration problems of polyominoes, several subclasses were defined by combining the two simple notions of convexity and directed growth. A polyomino is said to be column convex (resp. row convex) if every its column (resp. row) is connected (see Figure 1 (b)). A polyomino is said to be convex, if it is both row and column convex (see Figure 1 (c)). The area of a polyomino is just the number of cells it contains, while its semi-perimeter is half the number of edges of cells in its boundary. Thus, for any convex polyomino the semi-perimeter is the sum of the numbers of its rows and columns. Moreover, any convex polyomino is contained in a rectangle in the square lattice which has the same semi-perimeter, called minimal bounding rectangle.