TR#: | PHD-2011-11 |

Class: | PHD |

Title: | Enumeration of Lattice Animals |

Authors: | Gadi Aleksandrowicz |

Supervisors: | Gill Barequet |

PHD-2011-11.pdf | |

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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