The number of Z-convex polyominoes

Duchi, Enrica; Rinaldi, Simone; Schaeffer, Gilles

The number of Z-convex polyominoes

Authors

Duchi, Enrica

Rinaldi, Simone

Schaeffer, Gilles

Description

In this paper we consider a restricted class of convex 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 to write a system of functional equations for their generating functions. The generating function P(t) of Z-convex polyominoes with respect to the semi-perimeter turns out to be algebraic all the same and surprisingly, like the generating function of convex polyominoes, it can be expressed as a rational function of t and the generating function of Catalan numbers.
15 pages, 14 figures