Technical Report PHD-2011-11

Title: Enumeration of Lattice Animals
Authors: Gadi Aleksandrowicz
Supervisors: Gill Barequet
Abstract: Lattice animals, which can be defined as connected subgraphs of the dual graph of a given lattice, are well-known combinatorial objects, studied both in pure and recreational mathematics, and in various different fields, such a statistical physics and computational geometry. Of particular interest are lattice animals on the standard rectangular lattice (called \emph{polyominoes} in the two dimensional case and \emph{polycubes} for higher dimensions).

In this work, we concern ourselves with the problem of \emph{enumerating} lattice animals: computing, or estimating, how many lattice animals exist for a given type and size. This is a difficult combinatorial problem; even for the relatively simple case of the two-dimensional rectangular lattice not much is known, and other lattices are much less studied.

We investigate the problem in several directions:

\textbf{Enumeration algorithms}: We describe an algorithm for enumerating directly lattice animals on every possible lattice. Using a parallel version of the algorithm we counted many types of lattice animals and found many results never given in the literature so far.

\textbf{Analytical analysis}: We describe formulae for several sequences of lattice animals (specifically, some of the diagonals in a table of polycubes, arranged by size and dimension).

\textbf{Transfer-matrix methods}: We describe a method for computing algorithmically the generating function of lattice animals on a so-called ``twisted cylinder''. These animals are of special interest since there are fewer of them than regular polyominoes (of any size $n$), thus, their growth-rate limit is a lower bound to the growth-rate of polyominoes (whose computation is one of the main problems in the field).

\textbf{Bijection with permutations}: We describe a novel method of identifying polyominoes with permutations, and characterizing classes of polyominoes using forbidden permutation patterns.

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