The number of Z-convex polyominoes

Details The number of Z-convex polyominoes The number of Z-convex polyominoes

2006-02-07

arxiv.org/abs/math/0602124v1

Mathematics

Combinatorics

15 pages, 14 figures

Scientific paper

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.

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