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… 

Growth rates of permutation classes: from countable to uncountable

  • Vincent Vatter
  • Mathematics
    Proceedings 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

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

TLDR
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

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

TLDR
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.

References

SHOWING 1-10 OF 26 REFERENCES

Growth rates of permutation classes: from countable to uncountable

  • Vincent Vatter
  • Mathematics
    Proceedings 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

Intervals of Permutation Class Growth Rates

We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is θB ≈ 2:35526, and that it also contains every value at least θB ≈ 2:35698.

Small permutation classes

We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number κ, approximately 2.20557, for which there are

On Growth Rates of Closed Permutation Classes

A class of permutations is called closed if 2 implies 2 , where the relation is the natural containment of permutations. Let n be the set of all permutations of 1, 2 ,...,n belonging to . We

Inflations of geometric grid classes of permutations

Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main

PERMUTATION CLASSES OF EVERY GROWTH RATE ABOVE 2.48188

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

Excluded permutation matrices and the Stanley-Wilf conjecture

Overview of some general results in combinatorial enumeration

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

Restricted permutations, antichains, atomic classes and stack sorting

Involvement is a partial order on all finite permutations, of infinite dimension and having subsets isomorphic to every countable partial order with finite descending chains. It has attacted the

Restricted permutations