The combinatorics of convex permutominoes

A permutomino of size n is a polyomino determined by particular pairs of permutations of n. Here we study various classes of convex permutominoes. We determine some combinatorial properties and, in particular, the characterization for the permutations defining convex, directed-convex, and parallelogram permutominoes. Using standard combinatorial techniques we provide a recursive decomposition for permutations associated with convex permutominoes, and we derive a closed formula for the number of these permutations of size n.