New York Journal of Mathematics
Volume 7 (2001) 223-231


Doron Zeilberger

The Umbral Transfer-Matrix Method, III: Counting Animals

Published: November 9, 2001
Keywords: Lattice animals, polyominoes, umbral calculus, functional equations
Subject: 05A15, 05A40, 82-04

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.


Supported in part by the NSF. I thank the referee for numerous corrections and improvements

Author information

Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA