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An Elementary Proof of Bevan's Theorem on the Growth of Grid Classes of Permutations

Part of: Combinatorics

Published online by Cambridge University Press:  11 March 2019

Michael Albert  and
Vincent Vatter
Michael Albert
Affiliation:
Department of Computer Science, University of Otago, Dunedin, New Zealand
Vincent Vatter
Affiliation:
Department of Mathematics, University of Florida, Gainesville, FL, USA (vatter@ufl.edu)
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Abstract

Bevan established that the growth rate of a monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. We give an elementary and self-contained proof of a generalization of this result using only Stirling's formula, the method of Lagrange multipliers, and the singular value decomposition of matrices. Our proof relies on showing that the maximum over the space of n × n matrices with non-negative entries summing to one of a certain function of those entries, parametrized by the entries of another matrix Γ of non-negative real numbers, is equal to the square of the largest singular value of Γ and that the maximizing point can be expressed as a Hadamard product of Γ with the tensor product of singular vectors for its greatest singular value.

Keywords

grid classesgrowth ratespermutation patternssingular values

MSC classification

Primary: 05A05: Permutations, words, matrices
Secondary: 05A16: Asymptotic enumeration
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 975 - 984
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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References

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