How to bound Klarner's constant without (a huge number of) Klarner--Rivest twigs
Vuong Bui
TL;DR
The paper addresses the long-standing problem of bounding Klarner's constant $\lambda$, the exponential growth rate of polyominoes. It introduces a recurrence-based framework of interacting local neighborhoods (twigs) that replaces exhaustive trillions of configurations with a compact, verifiable one-page system of convolution-type recurrences. Using a simple positivity-lemma approach, it first achieves $\lambda \le 4.63$, then tightens to $\lambda \le 4.5238$ by expanding the neighborhood types and their interactions. The method is modular and extensible, offering a practical path to improved bounds for other combinatorial systems governed by local constraints.
Abstract
Although known lower bounds for the growth rate $λ$ 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 (``twigs'') yielded $λ\le 4.649551$, later refined by Barequet and Shalah (2022) to $λ\le 4.5252$ using trillions of configurations. The inefficiency lies in representing each polyomino as an almost unrestricted sequence of twigs once the large set of neighborhoods is fixed. We introduce a recurrence-based framework that constrains how local neighborhoods concatenate. Using a small system of convolution-type recurrences, we obtain $λ\le 4.5238$. The proof is short, self-contained, and fully verifiable by hand. Despite the marginal numerical improvement, the main contribution is methodological: replacing trillions of configurations with a concise one-page system of recurrences. The framework can be extended, with modest computational assistance, to further tighten the bound and to address other combinatorial systems governed by similar local constraints.