Enumeration Bounds via an Isoperimetric-Type Inequality

In 1949, Loomis and Whitney published a geometrically intuitive inequality that bounds the cardinality of a d-dimensional set in terms of the cardinalities of its projections onto the coordinate hyperplanes. We show how this inequality can be used to prove two results in the asymptotic enumeration o...

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Bibliographic Details
Published in:Journal of physics. Conference series 2006-06, Vol.42 (1), p.213-220
Main Author: Madras, N
Format: Article
Language:English
Subjects:
Animals
Cubic lattice
Enumeration
Hyperplanes
Inequality
Physics
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Summary:In 1949, Loomis and Whitney published a geometrically intuitive inequality that bounds the cardinality of a d-dimensional set in terms of the cardinalities of its projections onto the coordinate hyperplanes. We show how this inequality can be used to prove two results in the asymptotic enumeration of lattice animals: a bound on the critical exponent for the number of lattice animals in arbitrary dimension, and a bound on the growth constant for the number of almost unknotted embeddings of graphs in the cubic lattice.
ISSN:1742-6596
1742-6588
1742-6596
DOI:10.1088/1742-6596/42/1/018