Ascending Convex Polyominoes

TL;DR

This paper studies the structure and enumeration of Z-convex polyominoes and related subclasses, deriving explicit algebraic generating functions and asymptotic growth rates through combinatorial decompositions and functional equations.

Contribution

It introduces ascending polyominoes and a decomposition into subclasses, providing explicit algebraic generating functions and asymptotic analysis for Z-convex polyominoes.

Findings

Explicit algebraic generating functions for subclasses of Z-convex polyominoes.

Asymptotic growth rates of subclasses are determined.

Decomposition into three disjoint subclasses facilitates enumeration.

Abstract

Convex polyominoes can be refined according to the number of direction changes in monotone paths connecting pairs of cells, leading to the notion of $k$-convexity. In particular, the cases $k=1$ and $k=2$ correspond to $L$-convex and $Z$-convex polyominoes, two well-studied subclasses of convex polyominoes, with intermediate families such as centered and $4$-stack polyominoes. These families exhibit remarkably different combinatorial behaviours, suggesting that geometric constraints have a strong impact on the nature of the generating function: $L$-convex and centered polyominoes possess rational generating functions and growth of order $(2+\sqrt{2})^n$ and $4^n$, respectively, while $Z$-convex, 4-stack, and convex polyominoes have algebraic functions and asymptotics of order $n4^n$, $\sqrt{n}\,4^n$, and $n4^n$ respectively. In this paper we investigate the structure of $Z$-convex polyominoes by introducing a refinement based on the NW- and NE-convexity degrees, which yields a decomposition into three disjoint subclasses $C(1,2)$, $C(2,1)$, and $C(2,2)$. To enumerate these families we introduce ascending polyominoes, admitting a simple geometric characterization, and construct a generating tree that leads to functional equations for the corresponding generating functions. By solving these equations we obtain explicit algebraic generating functions and the asymptotic growth for all the subclasses.