ZOO

The umbral transfer-matrix method. III: Counting animals. par This is the third part of the five-part saga on the umbral transfer-matrix method, based on Gian-Carlo Rota’s seminal notion of the umbra. In this article we describe the Maple package ZOO that for any specific $k$, automatically constructs an umbral scheme for enumerating “$k$-board” lattice animals (polyominoes) on the two-dimensional square lattice. Such umbral schemes enable counting these important classes of animals in polynomial time as opposed to the exponential time that is required for counting all animals.