A Geometric Bijection for $xy$-Convex Curves and Convex Polyominoes

A connected subset of ${\mathbb R}^2$ consisting of unit squares with integral vertices is called a convex polyomino or is simply said to be $xy$-convex if it intersects any horizontal or vertical line exactly in one closed interval. In this paper, a geometric representation for xy-convex sets is described, allowing us to obtain, by elementary combinatorial methods, known formulas for the number of convex polyominoes contained in a rectangle of given size.