Abstract
We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of the number of polyominoes (connected subsets of grid squares) with a given number of squares. We achieve this by analyzing polyominoes on a different surface, a so-called $\textit{twisted cylinder}$ by the transfer matrix method. A bijective representation of the "states'' of partial solutions is crucial for allowing a compact representation of the successive iteration vectors for the transfer matrix method.
Highlights
A polyomino of size n, called an n-omino, is a connected set of n adjacent squares on a regular square lattice. (Connectivity is through edges only)
Fixed polyominoes are considered distinct if they have different shapes or orientations
The six fixed triominoes—polyominoes of size 3 are shown on the side
Summary
A polyomino of size n, called an n-omino, is a connected set of n adjacent squares on a regular square lattice. (Connectivity is through edges only). The most successful approach for computing values of A(n) is Jensen’s transfer matrix algorithm [4, 5], which is based on the algorithm of Conway and Guttmann [3]. The success of Jensen’s approach is based on treating only a small fraction of all possible states. This so-called pruning of states requires the algorithms to encode and store states explicitly, using a hash table. In this paper we count polyominoes on a different grid structure, a twisted cylinder. The twist allows us to build up the cylinder incrementally, one cell at a time, through a uniform process. The full paper [2] contains all details and proofs which are omitted in this extended abstract
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