Permutations Defining Convex Permutominoes

A permutomino of size n is a polyomino determined by particular pairs of permutations of size n. Here we determine the combinatorial properties and, in particular, the characterization for the pairs of 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. We provide a closed formula for the number of these permutations with size n.