Zeilberger, Doron The umbral transfer-matrix method. III: Counting animals. (English) Zbl 0987.05009 New York J. Math. 7, 223-231 (2001). Author’s abstract: 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. Reviewer: Ian Anderson (Glasgow) Cited in 4 Documents MSC: 05A15 Exact enumeration problems, generating functions 05A40 Umbral calculus 82-04 Software, source code, etc. for problems pertaining to statistical mechanics 05B50 Polyominoes Keywords:lattice animals; polyominoes; umbral calculus; functional equations; counting animals Software:Maple; MAYLIS; ZOO; SAW; SAP; ANIMALS; MARKOV; USAP; USAW; FreeANIMALS; FreeSAP; FreeSAW; ROTA PDFBibTeX XMLCite \textit{D. Zeilberger}, New York J. Math. 7, 223--231 (2001; Zbl 0987.05009) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: Number of board-pair-pile polyominoes with n cells.