An asymptotic lower bound on the number of polyominoes

TL;DR

This paper establishes an asymptotic lower bound on the number of polyominoes, advancing understanding of their growth rate and providing bounds on Klarner's constant using analytical techniques.

Contribution

It introduces a new asymptotic lower bound for polyomino counts and proposes conjectures relating to their ratio, aiding in bounding Klarner's constant.

Findings

Established a lower bound: P(n) ≥ An^{-T log n} λ^n

Under a conjecture, improved the bound to P(n) ≥ An^{-T} λ^n

Suggested an approach to bound λ from above, close to current bounds

Abstract

Let $P(n)$ be the number of polyominoes of $n$ cells and $\lambda$ be Klarner's constant, that is, $\lambda=\lim_{n\to\infty} \sqrt[n]{P(n)}$. We show that there exist some positive numbers $A,T$, so that for every $n$ \[ P(n) \ge An^{-T\log n} \lambda^n. \] This is somewhat a step toward the well known conjecture that there exist positive $C,\theta$ so that $P(n)\sim Cn^{-\theta}\lambda^n$ for every $n$. In fact, if we assume another popular conjecture that $P(n)/P(n-1)$ is increasing, we can get rid of $\log n$ to have \[ P(n)\ge An^{-T}\lambda^n. \] Beside the above theoretical result, we also conjecture that the ratio of the number of some class of polyominoes, namely inconstructible polyominoes, over $P(n)$ is decreasing, by observing this behavior for the available values. The conjecture opens a nice approach to bounding $\lambda$ from above, since if it is the case, we can conclude that \[ \lambda < 4.1141, \] which is quite close to the current best lower bound $\lambda > 4.0025$ and greatly improves the current best upper bound $\lambda < 4.5252$. The approach is merely analytically manipulating the known or likely properties of the function $P(n)$, instead of giving new insights of the structure of polyominoes. The techniques can be applied to other lattice animals and self-avoiding polygons of a given area with almost no change.