Asymptotics of Z-convex polyominoes

Abstract

The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper we show that one can compute in polynomial time the number of polyominoes of area n and degree of convexity at most 2 (the so-called Z-convex polyominoes). The integer sequence that we have computed allows us to conjecture the asymptotic number an of Z-convex polyominoes of area n, ɑn ∼ C·exp(π)√11n/4⁄n3/2.