Symmetric convex permutominoes and involutions

A permutomino of size n is a polyomino determined by particular pairs of permutations of length n. In this paper we consider the class of convex permutominoes which are symmetric with respect to the diagonal x = y. We determine the number of these permutominoes according to their size and we characterize the class of permutations associated to these objects as particular involutions of length n. To do this we need to introduce a larger class of objects, called symmetric permutominides, and to study their combinatorial properties