# Growth rates of permutation classes: Categorization up to the uncountability threshold

@article{Pantone2020GrowthRO,
title={Growth rates of permutation classes: Categorization up to the uncountability threshold},
author={Jay Pantone and Vincent Vatter},
journal={Israel Journal of Mathematics},
year={2020},
volume={236},
pages={1-43}
}
• Published 13 May 2016
• Mathematics
• Israel Journal of Mathematics
In the antecedent paper to this it was established that there is an algebraic number ξ ≈ 2.30522 such that while there are uncountably many growth rates of permutation classes arbitrarily close to ξ , there are only countably many less than ξ . Here we provide a complete characterization of the growth rates less than ξ . In particular, this classification establishes that ξ is the least accumulation point from above of growth rates and that all growth rates less than or equal to ξ are achieved…
5 Citations

### Growth rates of permutation classes: from countable to uncountable

• Vincent Vatter
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We establish that there is an algebraic number ξ≈2.30522 such that while there are uncountably many growth rates of permutation classes arbitrarily close to ξ , there are only countably many less

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It is proved that equality holds if the class is sum closed, and in the special case where the growth rate is at most $\xi \approx 2.30522$, using results from Pantone and Vatter on growth rates less than $\xi$.

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

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Proceedings of the Edinburgh Mathematical Society
• 2019
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

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• Vincent Vatter
• Mathematics
Proceedings of the London Mathematical Society
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We establish that there is an algebraic number ξ≈2.30522 such that while there are uncountably many growth rates of permutation classes arbitrarily close to ξ , there are only countably many less

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We prove that there are permutation classes (hereditary properties of permutations) of every growth rate (Stanley-Wilf limit) at least \lambda \approx 2.48187, the unique real root of

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This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include

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