A Closed Formula for the Number of Convex Permutominoes

Abstract

In this paper we determine a closed formula for the number of convex permutominoes of size $n$. We reach this goal by providing a recursive generation of all convex permutominoes of size $n+1$ from the objects of size $n$, according to the ECO method, and then translating this construction into a system of functional equations satisfied by the generating function of convex permutominoes. As a consequence we easily obtain also the enumeration of some classes of convex polyominoes, including stack and directed convex permutominoes.