Enumeration of 4-stack polyominoes

Abstract

In this paper, we consider the class of 4-stack polyominoes, i.e. polyominoes which can be decomposed into a central rectangle supporting four stack polyominoes, one on each side of the rectangle. This class of objects–recently introduced by Marc Noy–extends the class of centered convex polyominoes, and is included into the class of Z-convex polyominoes. Using an inclusion/exclusion approach, we obtain the enumeration of 4-stack polyominoes according to the number of rows and columns. Moreover, we solve some problems posed by Marc Noy, proving that the generating function of 4-stack polyominoes according to the semi-perimeter is algebraic, and that their asymptotic behavior is $\sqrt{n}{4}^n$, which is immediately smaller than the asymptotic behavior of Z-convex polyominoes, which is $n{4}^n$. As a corollary of our result, we find the generating function of bi-centered polyominoes, i.e. convex polyominoes which are both horizontally and vertically centered.