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

Published online by Cambridge University Press:  11 March 2019

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)

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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References

1Albert, M., Pantone, J. and Vatter, V., On the growth of merges and staircases of permutation classes, Rocky Mountain J. Math., to appear. arXiv:1608.06969.Google Scholar
2Bevan, D., Growth rates of permutation grid classes, tours on graphs, and the spectral radius, Trans. Amer. Math. Soc. 367 (2015), 58635889.CrossRefGoogle Scholar
3Bevan, D., On the growth of permutation classes, PhD thesis, Open University (2015).Google Scholar
4Marcus, A. and Tardos, G., Excluded permutation matrices and the Stanley–Wilf conjecture, J. Combin. Theory Ser. A 107 (2004), 153160.CrossRefGoogle Scholar
5Pantone, J. and Vatter, V., Growth rates of permutation classes: categorization up to the uncountability threshold, Israel J. Math., to appear. arXiv:1605.04289.Google Scholar
6Stankova, Z., Forbidden subsequences, Discrete Math. 132 (1994), 291316.CrossRefGoogle Scholar
7Vatter, V., Growth rates of permutation classes: from countable to uncountable, Proc. Lond. Math. Soc. (3), to appear. arXiv:1605.04297.Google Scholar
8Vatter, V., Small permutation classes, Proc. Lond. Math. Soc. (3) 103 (2011), 879921.CrossRefGoogle Scholar
9Vatter, V., Permutation classes, In Handbook of enumerative combinatorics (ed. Bóna, M.), pp. 754833 (CRC Press, Boca Raton, FL, 2015).Google Scholar

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