# 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} }

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

- MathematicsProceedings of the London Mathematical Society
- 2019

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…

### Two examples of Wilf-collapse

- MathematicsDiscrete Mathematics & Theoretical Computer Science
- 2021

Two permutation classes, the X-class and subpermutations of the increasing
oscillation are shown to exhibit an exponential Wilf-collapse. This means that
the number of distinct enumerations of…

### On the centrosymmetric permutations in a class

- MathematicsAustralas. J Comb.
- 2019

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

- MathematicsProceedings 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…

### Packing and Counting Permutations

- Mathematics, Computer Science
- 2018

The main tool for the upper bounds is the framework of flag algebras introduced by Razborov in 2007, and this work presents Permpack — a flag algebra package for permutations.

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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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