Haruspicy and anisotropic generating functions
type: Item
instance of: scholarly article
title: Haruspicy and anisotropic generating functions
zbMATH Open document ID: 1023.05031
DOI: 10.1016/S0196-8858(02)00534-1
author: Andrew Rechnitzer
published in: Advances in Applied Mathematics
publication date: 2003-07-29T00:00:00Z
review text: The anisotropic generating function for a family of bond animals uses two variables; \(x\) to count the horizontal bonds and \(y\) to count the vertical bonds. The author shows the following results for the anisotropic generating function of some general classes of bond animals. The coefficient of \(y^n\) has the form \(P_n(x)/D_n(x)\) where \(D_n(x)\) is a cyclotomic polynomial and \(P_n(x)\) is a polynomial whose degree does not exceed the degree of \(D_n(x)\). An explicit polynomial \(B_n(x)\) is given such that \(D_n(x)\) necessarily divides \(B_n(x)\). Consequently \(P_n(x)/D_n(x)\) can be computed exactly from the first \(O(n^3)\) terms of its expansion. The author has done an excellent job of presenting his results in an accessible way and of explaining their wider context. This is a very readable paper. The results are proved by a technique which the author calls `haruspicy' in a humorous reference to the ancient art of divination by the disection of animals. On the basis of numerical evidence and of comparison with solved families the author speculates (a) that his families of bond animals are not solvable and (b) that \(B_n(x)\) is a good upper bound for \(D_n(x)\) in the sense that the degree of \(D_n(x)/B_n(x)\) is small (in some imprecise sense).
Mathematics Subject Classification ID: 05B50
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 1956069
zbMATH Keywords: bond animal
zbMATH Keywords: square lattice
zbMATH Keywords: cyclotomic polynomial
zbMATH Keywords: anisotropic generating function
zbMATH Keywords: differentiably finite
reviewed by: Ian M. Wanless
MaRDI profile type: Publication
cites work: A method for the enumeration of various classes of column-convex polygons
cites work: Enumeration of three-dimensional convex polygons
cites work: Inversion relations, reciprocity and polyominoes
cites work: The number of three-choice polygons
cites work: On two-dimensional percolation
cites work: Indicators of solvability for lattice models
cites work: Solvability of Some Statistical Mechanical Systems
cites work: On the symmetry classes of planar self-avoiding walks
cites work: Scaling prediction for self-avoiding polygons revisited
cites work: Cell Growth Problems
cites work: D-finite power series
cites work: On an error term involving the totient function
cites work: Haruspicy 2: The anisotropic generating function of self-avoiding polygons is not D-finite
cites work: Some problems in the counting of lattice animals, polyominoes, polygons and walks
cites work: On three-dimensional self-avoiding walk symmetry classes
cites work: Differentiably finite power series
cites work: Q5781292
description: scientific article; zbMATH DE number 1956069