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Growth rates of geometric grid classes of permutations [article]

David Bevan
2014 arXiv   pre-print
Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes.  ...  By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric grid class has a growth rate which is given by the square of the largest root of the  ...  Then, we explore the influence of cycle parity on growth rates and re- late the growth rates of geometric grid classes to those of monotone grid classes.  ... 
arXiv:1306.4246v2 fatcat:tq5skzzqfbaajfhjci2suxlhda

Growth Rates of Geometric Grid Classes of Permutations

David Bevan
2014 Electronic Journal of Combinatorics  
Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes.  ...  By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric grid class has a growth rate which is given by the square of the largest root of the  ...  S.D.G. the electronic journal of combinatorics 21(4) (2014), #P4.51  ... 
doi:10.37236/4834 fatcat:wuu2bxkbsvfsff2eqtext25isi

Growth rates of geometric grid classes of permutations

David Bevan
unpublished
Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes.  ...  By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric grid class has a growth rate which is given by the square of the largest root of the  ...  S.D.G. the electronic journal of combinatorics 21(4) (2014), #P4.51  ... 
fatcat:wtqct6qr5zg2pln6mejl7n55fq

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

Vincent Vatter
2019 arXiv   pre-print
Central to the proof are various structural notions regarding generalized grid classes and a new property of permutation classes called concentration.  ...  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  ...  1 Every upper growth rate of a permutation class is the growth rate of a sum closed class.  ... 
arXiv:1605.04297v3 fatcat:fbvrvghpzzdepo4geb42yipmmm

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 Grid(M ) is equal to the greatest upper growth rate of the monotone grid class of a connected component of M .  ... 
arXiv:1409.5159v3 fatcat:epgv2blhfjd6flmjcdqmf66dka

On the centrosymmetric permutations in a class [article]

Justin M. Troyka
2019 arXiv   pre-print
We prove one direction of inequality for sum closed classes and for some geometric grid classes.  ...  Initially motivated by a question by Alexander Woo, we investigate the question of whether the growth rate of a permutation class equals the growth rate of its even-size centrosymmetric elements.  ...  I also thank Michael Albert, David Bevan, and Robert Brignall (and a few others attending Permutation Patterns 2018) for pointing out a very false "theorem" that was in an earlier draft of this paper.  ... 
arXiv:1804.03686v3 fatcat:ldifmvj5cfdntpfgfscw6udxei

Inflations of geometric grid classes of permutations

Michael H. Albert, Nik Ruškuc, Vincent Vatter
2014 Israel Journal of Mathematics  
Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes.  ...  This bound is tight as there are permutation classes with growth rate κ which have nonrational generating functions.  ...  classes of growth rate less than κ « 2.20557.  ... 
doi:10.1007/s11856-014-1098-8 fatcat:incyrf4zivfudpgb5pgtior6i4

Inflations of geometric grid classes of permutations [article]

Michael H. Albert, Nik Ruskuc, Vincent Vatter
2012 arXiv   pre-print
Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes.  ...  This bound is tight as there are permutation classes with growth rate κ which have nonrational generating functions.  ...  classes of growth rate less than κ « 2.20557.  ... 
arXiv:1202.1833v1 fatcat:cvefgopei5ctvlvmyyppbt46y4

On the growth of permutation classes [article]

David Bevan
2015 arXiv   pre-print
We characterise the growth rates of geometric grid classes in terms of the spectral radii of trees.  ...  We prove that the growth rate of a grid class is given by the square of the spectral radius of an associated graph and deduce some facts relating to the set of grid class growth rates.  ...  Then, we explore the influence of cycle parity on growth rates and relate the growth rates of geometric grid classes to those of monotone grid classes.  ... 
arXiv:1506.06688v1 fatcat:jgznt4crvrdt5dcjqiaelwdo5m

Growth rates of permutation grid classes, tours on graphs, and the spectral radius

David Bevan
2015 Transactions of the American Mathematical Society  
Associated with grid class Grid(M) is a graph, G(M), known as its "row-column" graph. We prove that the exponential growth rate of Grid(M) is equal to the square of the spectral radius of G(M).  ...  Monotone grid classes of permutations have proven very effective in helping to determine structural and enumerative properties of classical permutation pattern classes.  ...  Acknowledgements Grateful thanks are due to Robert Brignall for numerous discussions related to this work, and for much helpful advice and thorough feedback on earlier drafts of this paper, and also to  ... 
doi:10.1090/s0002-9947-2015-06280-1 fatcat:v4i2hu2owbcojanikjntjxbxvu

Geometric grid classes of permutations [article]

Michael H. Albert, M. D. Atkinson, Mathilde Bouvel, Nik Ruškuc,, Vincent Vatter
2012 arXiv   pre-print
A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope \pm1 arranged in a rectangular pattern governed by a matrix.  ...  Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating  ...  the possible growth rates of permutation classes, and • to provide necessary and sufficient conditions for permutation classes to have amenable generating functions.  ... 
arXiv:1108.6319v2 fatcat:ayoisy7m3bclzpre5js4byehva

Geometric grid classes of permutations

Michael H. Albert, M. D. Atkinson, Mathilde Bouvel, Nik Ruškuc, Vincent Vatter
2013 Transactions of the American Mathematical Society  
A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ±1 arranged in a rectangular pattern governed by a matrix.  ...  Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating  ...  the possible growth rates of permutation classes and • to provide necessary and sufficient conditions for permutation classes to have amenable generating functions.  ... 
doi:10.1090/s0002-9947-2013-05804-7 fatcat:oqqn3ygkzngohe4hucg2b5x7fi

Geometric morphometrics of different malocclusions in lateral skull radiographs

Josef Freudenthaler, Aleš Čelar, Christopher Ritt, Philipp Mitteröcker
2016 Journal of Orofacial Orthopedics  
grid visualization, permutation tests, and receiver operating characteristic curves.  ...  To evaluate the role of craniofacial shape in malocclusion by application of geometric morphometrics to a set of two-dimensional landmarks and semilandmarks obtained from lateral skull radiographs.  ...  This unique divergence from the growth of the class I group may represent unbalanced growth rates or remodelling of craniofacial structures in class III individuals.  ... 
doi:10.1007/s00056-016-0057-x pmid:27796401 pmcid:PMC5247554 fatcat:ua4rzbi3ijen5cokp62vukkfb4

Thin-plate spline analysis of mandibular growth

L Franchi, T Baccetti, J A McNamara
2001 Angle Orthodontist  
Centroid size was used as the measure of the geometric size of each mandibular specimen. Differences in size at the 6 developmental phases were tested statistically.  ...  Differences in shape between average mandibular configurations at the 6 developmental stages were visualized by means of thin-plate spline analysis and subjected to permutation test.  ...  The growth rate of craniofacial skeletal structures such as the mandible is not linear during development.  ... 
doi:10.1043/0003-3219(2001)071<0083:tpsaom>2.0.co;2 pmid:11302593 fatcat:xcv2z2zomjapvbxslnvn32wymi

Combinatorial Exploration: An algorithmic framework for enumeration [article]

Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, Henning Ulfarsson
2022 arXiv   pre-print
We then apply Combinatorial Exploration to the domain of permutation patterns, to great effect. We rederive hundreds of results in the literature in a uniform manner and prove many new ones.  ...  These results can be found in a new public database, the Permutation Pattern Avoidance Library (PermPAL) at https://permpal.com.  ...  Gridded permutations Murphy and Vatter [121] introduced the notion of grid classes as a means of defining one permutation class as a geometric combination of others.  ... 
arXiv:2202.07715v1 fatcat:3iutygbjcnc4jbe4kkuvbsyo5m
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