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The umbral transfer-matrix method. III: Counting animals. (English) Zbl 0987.05009

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.

MSC:

05A15 Exact enumeration problems, generating functions
05A40 Umbral calculus
82-04 Software, source code, etc. for problems pertaining to statistical mechanics
05B50 Polyominoes
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Online Encyclopedia of Integer Sequences:

Number of board-pair-pile polyominoes with n cells.