An Asymptotic Lower Bound on the Number of Polyominoes

Abstract

Let P(n) be the number of polyominoes of n cells and \(\lambda \) be Klarner’s constant, that is, \(\lambda =\lim _{n\rightarrow \infty } \root n \of {P(n)}\). We show that there exist some positive numbers A, T, so that for every n

$$\begin{aligned} P(n) \ge An^{-T\log n} \lambda ^n. \end{aligned}$$

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

$$\begin{aligned} P(n)\ge An^{-T}\lambda ^n. \end{aligned}$$

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

$$\begin{aligned} \lambda < 4.1141, \end{aligned}$$

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.