A convolutional approach to bounding the number of polyominoes

TL;DR

This paper introduces a convolutional recurrence approach to improve upper bounds on the growth rate of polyominoes, providing a concise, verifiable method that surpasses previous trillion-configuration techniques.

Contribution

The paper presents a novel recurrence-based method that simplifies bounding polyomino growth, replacing extensive enumeration with a small, self-contained system of recurrences.

Findings

Achieved an upper bound of λ ≤ 4.5238 on polyomino growth rate.

Developed a new technique for rigorously bounding recurrences with a small parameter set.

Provided a short, self-contained proof that is easy to verify.

Abstract

Although known lower bounds for the growth rate $\lambda$ of polyominoes, or Klarner's constant, are already close to the empirically estimated value $4.06$, almost no conceptual progress on upper bounds has occurred since the seminal work of Klarner and Rivest (1973). Their approach, based on enumerating millions of local neighborhoods (also called ``twigs'') yielded $\lambda \le 4.649551$, later refined by Barequet and Shalah (2022) to $\lambda \le 4.5252$ using trillions of configurations. The inefficiency lies in representing each polyomino as an almost unrestricted sequence of neighborhoods once the large set of neighborhoods is fixed. We introduce a recurrence-based approach that constrains how local neighborhoods concatenate. Using a small system of convolution-type recurrences, we obtain $\lambda \le 4.5238$. The proof is short, self-contained, and hand-checkable. Despite the marginal numerical improvement, the main contribution is methodological: replacing trillions of configurations with a concise one-page system of recurrences. In addition, we present a new technique for rigorously bounding the growth of recurrences to any precision, applicable to a broad range of settings with nonnegative coefficients. The resulting upper bound even comes with a nice feature: a small set of parameters serves as the certificate for the bound, that is, one does not need to check more than a few arithmetic calculations to trust the bound.