The number of directed k-convex polyominoes

TL;DR

This paper introduces a new method to derive generating functions for directed convex polyominoes, specifically focusing on directed k-convex polyominoes, revealing their rational generating functions and asymptotic properties.

Contribution

The paper develops a novel method to compute generating functions for directed convex polyominoes, including the k-convex subclass, and analyzes their asymptotic behavior.

Findings

Generating function for directed k-convex polyominoes is rational.

Method applies to various families of directed convex polyominoes.

Asymptotic analysis of the generating function is provided.

Abstract

We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed k-convex polyominoes. We show it is a rational function and we study its asymptotic behavior.