The number of Z-convex polyominoes

In this paper we consider a restricted class of polyominoes that we call Z-convex polyominoes. Z-convex polyominoes are polyominoes such that any two pairs of cells can be connected by a monotone path making at most two turns (like the letter Z). In particular they are convex polyominoes, but they appear to resist standard decompositions. We propose a construction by "inflation" that allows us, through a quite tedious case analysis, to write a system of functional equations for their generating functions. Even though intermediate steps involve heavy computations, it turns out in the end that the generating function P(t) of Z-convex polyominoes with respect to the semi-perimeter can be expressed as a simple rational function of t and the generating function of Catalan numbers, like the generating function of convex polyominoes.