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Intervals of permutation class growth rates [article]

David Bevan
2015 arXiv   pre-print
We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is θ_B≈2.35526, and that it also contains every value at least λ_B≈2.35698.  ...  These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least λ_A≈2.48187.  ...  He would also like to thank Vince, Robert Brignall and two referees for reading earlier drafts of this paper; their feedback resulted in significant improvements to its presentation. S.D.G.  ... 
arXiv:1410.3679v2 fatcat:7i5poymeqvg35dyjv3pcbj7rym

Intervals of Permutation Class Growth Rates

David Bevan
2017 Combinatorica  
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 λB ≈ 2.35698.  ...  These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least λA ≈ 2.48187.  ...  He would also like to thank Vince, Robert Brignall and two referees for reading earlier drafts of this paper; their feedback resulted in significant improvements to its presentation. Soli Deo gloria!  ... 
doi:10.1007/s00493-016-3349-2 fatcat:fzh4oiwozrcvrpfgopkoc3enni

PERMUTATION CLASSES OF EVERY GROWTH RATE ABOVE 2.48188

Vincent Vatter
2009 Mathematika  
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 x^5-2x^4-2x^2-2x-  ...  1, thereby establishing a conjecture of Albert and Linton.  ...  permutation classes of growth rate κ.  ... 
doi:10.1112/s0025579309000503 fatcat:hk73x5eixzb2fdaahin4wotc7a

Large infinite antichains of permutations [article]

Michael H. Albert, Robert Brignall, Vincent Vatter
2012 arXiv   pre-print
We prove the existence and detail the construction of infinite antichains with arbitrarily large growth rates.  ...  Infinite antichains of permutations have long been used to construct interesting permutation classes and counterexamples.  ...  was stronger than the statement that every upper growth rate of a permutation class is algebraic.  ... 
arXiv:1212.3346v2 fatcat:ep5aggpnb5filaugpghfbjazm4

Growth rates of permutation classes: categorization up to the uncountability threshold [article]

Jay Pantone, Vincent Vatter
2019 arXiv   pre-print
We conclude by refuting a suggestion of Klazar, showing that ξ is an accumulation point from above of growth rates of finitely based permutation classes.  ...  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  ...  There are uncountably many permutation classes of growth rate κ.  ... 
arXiv:1605.04289v3 fatcat:fj5tomw7jfdfdmaxed4iryq7u4

Permutation classes [article]

Vincent Vatter
2015 arXiv   pre-print
This is a survey on permutation classes for the upcoming book Handbook of Enumerative Combinatorics.  ...  This chapter has greatly benefited by the comments, corrections, and suggestions of the referee as well as those of Michael Albert, David Bevan, Jonathan Bloom, Robert Brignall, Cheyne Homberger, Vít Jelínek  ...  The upper growth rate of a wpo permutation class is equal to the greatest upper growth rate of its atomic subclasses.  ... 
arXiv:1409.5159v3 fatcat:epgv2blhfjd6flmjcdqmf66dka

Some open problems on permutation patterns [article]

Einar Steingrimsson
2013 arXiv   pre-print
Other subjects treated are the Möbius function, topological properties and other algebraic aspects of the poset of permutations, ordered by containment, and also the study of growth rates of permutation  ...  classes, which are containment closed subsets of this poset.  ...  Acknowledgements I am deeply grateful to Vince Vatter, who provided invaluable help, in particular with the section on growth rates.  ... 
arXiv:1210.7320v2 fatcat:stgw6rpxerbtbdztalhbp3ft2q

Permutation patterns: basic definitions and notation [article]

David Bevan
2015 arXiv   pre-print
A brief presentation of basic definitions and notation used in permutation patterns research.  ...  A generating function is rational if it is the ratio of two polynomials. A generating function F(z) is algebraic if it can be defined as the root of a polynomial equation.  ...  Hence, the upper growth rate and lower growth rate of a class C are defined to be gr(C) = lim sup n→∞ |C n | 1/n and gr(C) = lim inf n→∞ |C n | 1/n .  ... 
arXiv:1506.06673v1 fatcat:4oheockukrerflrqn32hjm5mvm

Growth rates of permutation classes: from countable to uncountable [article]

Vincent Vatter
2019 arXiv   pre-print
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 than  ...  The classification of growth rates up to ξ is completed in a subsequent paper.  ...  1 Every upper growth rate of a permutation class is the growth rate of a sum closed class.  ... 
arXiv:1605.04297v3 fatcat:fbvrvghpzzdepo4geb42yipmmm

Growth rates for subclasses of Av(321) [article]

M. H. Albert, M. D. Atkinson, R. Brignall, N. Ruskuc, Rebecca Smith, J. West
2009 arXiv   pre-print
Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns  ...  The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth  ...  For the remainder of this paper we will only be using the second of these estimates; that the growth rate of a bounded merge of two permutation classes is the maximum of their individual growth rates.  ... 
arXiv:0903.1999v2 fatcat:csjjzq7mang6falfeay2g6536m

Growth Rates for Subclasses of $\mathrm{Av}(321)$

M. H. Albert, M. D. Atkinson, R. Brignall, N. Ruškuc, Rebecca Smith, J. West
2010 Electronic Journal of Combinatorics  
Pattern classes which avoid $321$ and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns  ...  The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth  ...  Growth rates offer a way of approximating the number of permutations of a given length in a pattern class.  ... 
doi:10.37236/413 fatcat:gifr6gaacrbqhd4tpbiazlejku

On Permutation Pattern Classes with Two Restrictions Only

M. D. Atkinson
2007 Annals of Combinatorics  
Permutation pattern classes that are defined by avoiding two permutations only and which contain only finitely many simple permutations are characterized and their growth rates are determined.  ...  For this reason K is called the growth rate of the class.  ...  In this paper we shall show how to find the growth rate of a 3-parameter family of pattern classes.  ... 
doi:10.1007/s00026-007-0320-3 fatcat:iaome2wbxnd53mmbxql6ola5fe

Growing at a Perfect Speed

M. H. ALBERT, S. A. LINTON
2009 Combinatorics, probability & computing  
and sal@dcs.st-and.ac.uk A collection of permutation classes is exhibited whose growth rates form a perfect set, thereby refuting some conjectures of Balogh, Bollobás and Morris.  ...  The set of growth rates of permutation classes includes some interval (λ, ∞). In fact, we suspect that the least such λ, if it exists, is not very large.  ...  A simple maximization exercise then shows that the growth rate of this new class is equal to the sum of the growth rates of A and of B.  ... 
doi:10.1017/s0963548309009699 fatcat:uxa5niy33nf47exhftfec23kuu

Small permutation classes

Vincent Vatter
2011 Proceedings of the London Mathematical Society  
countably many permutation classes of growth rate (Stanley-Wilf limit) less than κ but uncountably many permutation classes of growth rate κ, answering a question of Klazar.  ...  We go on to completely characterize the possible sub-κ growth rates of permutation classes, answering a question of Kaiser and Klazar.  ...  Is there a (upper, lower, proper) growth rate of a permutation class that is not achieved by a sum closed permutation class? The existence of growth rates.  ... 
doi:10.1112/plms/pdr017 fatcat:3vqwu775tzd27kzfoo4mvrpbfi

Inflations of geometric grid classes of permutations

Michael H. Albert, Nik Ruškuc, Vincent Vatter
2014 Israel Journal of Mathematics  
This bound is tight as there are permutation classes with growth rate κ which have nonrational generating functions.  ...  In particular, these are the two main tools in the recent classification of permutation classes of growth rate less than κ « 2.20557 (a specific algebraic integer at which infinite antichains first appear  ...  classes of growth rate less than κ « 2.20557.  ... 
doi:10.1007/s11856-014-1098-8 fatcat:incyrf4zivfudpgb5pgtior6i4
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