A Lattice Animal is a set of edge-connected cells in a given lattice.
For example, the Tetris game is played with Lattice Animals
with 4 cells in the two-dimensional orthogonal lattice. The
enumeration of Lattice Animals is a long-time standing problem,
arising in all of recreational mathematics, discrete geometry,
and statistical physics. In this talk I will discuss a
generalization of Redelmeier's algorithm to the enumeration of
polyominoes that lie on any structural (repetitve) lattice. I
will also present a bijection between polyominoes and
permutations, and describe the enumeration of several families
of polyominoes through couting classes of permutations that
avoid some sets of "forbidden patterns". In addition, I will
apply a transfer-matrix method to polyominoes on a twisted
cylinder in order to derive the generating functions of the
sequence that enumerates these polyominoes.
