# Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

## TL;DR

This paper improves upper bounds on the growth constants of polyominoes and polycubes, providing new theoretical bounds and computational results that advance understanding of their asymptotic behavior.

## Contribution

It introduces novel bounds on the growth constants for polyominoes and polycubes, including the first in nearly 50 years for two dimensions, and extends these bounds to higher dimensions using new generating function techniques.

## Key findings

- Proved that  b7 4.5252
- Established  b7 (2d-2)e + o(1) for all d b7 2
- Improved the upper bound for  from 12.2071 to 9.8073, and further to 9.3835 with iterative methods

## Abstract

A $d$-dimensional polycube is a facet-connected set of cells (cubes) on the $d$-dimensional cubical lattice $\mathbb{Z}^d$. Let $A_d(n)$ denote the number of $d$-dimensional polycubes (distinct up to translations) with $n$ cubes, and $\lambda_d$ denote the limit of the ratio $A_d(n{+}1)/A_d(n)$ as $n \to \infty$. The exact value of $\lambda_d$ is still unknown rigorously for any dimension $d \geq 2$; the asymptotics of $\lambda_d$, as $d \to \infty$, also remained elusive as of today. In this paper, we revisit and extend the approach presented by Klarner and Rivest in 1973 to bound $A_2(n)$ from above. Our contributions are: Using available computing power, we prove that $\lambda_2 \leq 4.5252$. This is the first improvement of the upper bound on $\lambda_2$ in almost half a century; We prove that $\lambda_d \leq (2d-2)e+o(1)$ for any value of $d \geq 2$, using a novel construction of a rational generating function which dominates that of the sequence $\left(A_d(n)\right)$; For $d=3$, this provides a subtantial improvement of the upper bound on $\lambda_3$ from 12.2071 to 9.8073; However, we implement an iterative process in three dimensions, which improves further the upper bound on $\lambda_3$to $9.3835$.