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On types of growth for graph-different permutations [article]

János Körner, Gábor Simonyi, Blerina Sinaimeri
2007 arXiv   pre-print
We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices.  ...  We give estimates for other cases and compare the results in case of complementary graphs.  ...  graphs G contains the Shannon capacity problem of all such graphs for which Shannon capacity is attained as the capacity within a type for some rational distribution.  ... 
arXiv:0712.1442v1 fatcat:3sbfjdwugbd3dlz6molpjvs6ge

On types of growth for graph-different permutations

János Körner, Gábor Simonyi, Blerina Sinaimeri
2009 Journal of combinatorial theory. Series A  
We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices.  ...  We give estimates for other cases and compare the results in case of complementary graphs.  ...  Acknowledgment Part of this research was done during a visit of the second author in Rome, where he enjoyed hospitality of Dipartimento di Informatica, Università "La Sapienza."  ... 
doi:10.1016/j.jcta.2008.11.003 fatcat:edtatfzjgjf4lb4iqtbvo4usry

On the permutation capacity of digraphs [article]

Gerard Cohen, Emanuela Fachini, Janos Korner
2008 arXiv   pre-print
We extend several results of the third author and C. Malvenuto on graph-different permutations to the case of directed graphs and introduce new open problems.  ...  Permutation capacity is a natural extension of Sperner capacity from finite directed graphs to infinite digraphs.  ...  We write N(G, n) for the largest cardinality of a set of pairwise G-different permutations of [n] . It is easy to see (cf.  ... 
arXiv:0809.1522v1 fatcat:djrpssavvbg5xckwbriscjx4ka

Substitutional subshifts and growth of groups [article]

Volodymyr Nekrashevych
2020 arXiv   pre-print
We show how to use symbolic dynamics of Schreier graphs to embed the Grigorchuk group into a simple torsion group of intermediate growth and to construct uncountably many growth types of simple torsion  ...  Since a finite index subgroup has the same growth type as the group, it follows that there is a continuum of different growth types of simple finitely generated groups.  ...  Our second result is showing that there are uncountably many pairwise different growth types among finitely generated simple groups.  ... 
arXiv:2008.04983v1 fatcat:6b7ltcbyirhl7aohjbvbxgrlka

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

Jay Pantone, Vincent Vatter
2019 arXiv   pre-print
A significant part of this classification is achieved via a reconstruction result for sum indecomposable permutations.  ...  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.  ...  We are grateful to David Bevan for his many insightful comments on this work. We are also grateful to the referees for their helpful suggestions.  ... 
arXiv:1605.04289v3 fatcat:fj5tomw7jfdfdmaxed4iryq7u4

On Growth Rates of Permutations, Set Partitions, Ordered Graphs and Other Objects

Martin Klazar
2008 Electronic Journal of Combinatorics  
For classes ${\cal O}$ of structures on finite linear orders (permutations, ordered graphs etc.) endowed with containment order $\preceq$ (containment of permutations, subgraph relation etc.), we investigate  ...  We present a framework of edge $P$-colored complete graphs $({\cal C}(P),\preceq)$ which includes many of these situations, and we prove for it two such restrictions (jumps in growth): $f(n)$ is eventually  ...  In this article we present a general framework for proving restrictions of the type 1-4 on growths of other classes of structures besides permutations and ordered graphs.  ... 
doi:10.37236/799 fatcat:4nbfuxm3fzgopaxrppoccazdyy

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.  ...  The containment order on permutations is much more similar to the induced subgraph order on graphs, and under this order graphs clearly do contain infinite antichains; one such antichain consists of all  ... 
arXiv:1212.3346v2 fatcat:ep5aggpnb5filaugpghfbjazm4

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.  ...  I am grateful to Michael Albert, David Bevan, Robert Brignall, Michael Engen, Jay Pantone, and the anonymous referees for their many insightful comments on this work.  ... 
arXiv:1605.04297v3 fatcat:fbvrvghpzzdepo4geb42yipmmm

On growth rates of permutations, set partitions, ordered graphs and other objects [article]

Martin Klazar
2007 arXiv   pre-print
For classes O of structures on finite linear orders (permutations, ordered graphs etc.) endowed with containment order cont (containment of permutations, subgraph relation etc.), we investigate restrictions  ...  We present a framework of edge P-colored complete graphs (C(P), cont) which includes many of these situations, and we prove for it two such restrictions (jumps in growth): f(n) is eventually constant or  ...  My thanks go to Toufik Mansour and Alek Vainshtein for their  ... 
arXiv:math/0703047v1 fatcat:p7hink2hdzg5vmxopcn7a7jh2u

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.  ...  For example, we could allow juxtapositions of permutations of three types in an L shape.  ...  The simplest example of this type is the class of permutations having at most one descent, which is the juxtaposition of the increasing class with itself.  ... 
doi:10.1017/s0963548309009699 fatcat:uxa5niy33nf47exhftfec23kuu

A graph-theoretic approach to testing associations between disparate sources of functional genomics data

R. Balasubramanian, T. LaFramboise, D. Scholtens, R. Gentleman
2004 Bioinformatics  
The congruency of these different data sources, or lack thereof, can shed light on the mechanisms that govern cellular function.  ...  permutation and node label permutation tests.  ...  A fitness score was computed for each gene-knockout strain in each medium based on the decrease in growth rate as compared to the wild-type strain in the same medium.  ... 
doi:10.1093/bioinformatics/bth405 pmid:15256415 fatcat:y6ywvj3hxvbnzlg4z27ggz6hza

Oligomorphic Permutation Groups [chapter]

Peter J. Cameron
2009 Perspectives in Mathematical Sciences II  
A permutation group G (acting on a set Ω, usually infinite) is said to be oligomorphic if G has only finitely many orbits on Ω n (the set of n-tuples of elements of Ω).  ...  We give a spread of examples, describe results on the growth rate of the counting functions, discuss a graded algebra associated with an oligomorphic group, and finally discuss group-theoretic properties  ...  For the automorphism group of the random graph, the growth rate is about exp(cn 2 ). For such growth, it doesn't make a lot of difference whether we consider F n or f n .  ... 
doi:10.1142/9789814273657_0003 fatcat:kxt2dwfkpvc45kfjwjkb2j6qma

Pairwise colliding permutations and the capacity of infinite graphs

János Körner, Claudia Malvenuto
2006 SIAM Journal on Discrete Mathematics  
We call two permutations of the first n naturals colliding if they map at least one number to consecutive naturals.  ...  We give bounds for the exponential asymptotics of the largest cardinality of any set of pairwise colliding permutations of [n].  ...  We would like to thank Miki Simonovits for his friendly interest.  ... 
doi:10.1137/050632877 fatcat:ajl25wgrmrcqdckowsg5cs3cki

A Recursive Algorithmic Approach to the Finding of Permutations for the Combination of Any Two Sets [article]

Diego Fernando C. Carrión L
2014 arXiv   pre-print
In this paper I present a conjecture for a recursive algorithm that finds each permutation of combining two sets of objects (AKA the Shuffle Product).  ...  The routes taken to find each of the permutations then form a series of associations or adjacencies which can be represented in a tree graph which appears to possess some properties of a fractal.  ...  This graph was used in the analysis of patterns which were emerging as we tried different methods to manipulate the data in search of the next permutation.  ... 
arXiv:1401.1450v1 fatcat:s7u3bmyhvne4lklvsv57bterxu

Economic and Social Institutions: Modelling the Evolution Paths for the Archaic Society

Vyacheslav Shvedovsky, Anton Standrik, Yuriy Bilan
2016 Economics & Sociology  
We use mapping of permutations to graphs for demonstrating that the evolution of society can be represented as a sequence of nested sub-groups in group lattices.  ...  Evolution paths and their complexity can be evaluated using the graph theory and the complexity criteria.  ...  initial graph for permutation has only one connection component (see the Appendix).  ... 
doi:10.14254/2071-789x.2016/9-2/9 fatcat:vl42ps3asjclfl4j4qvexgnya4
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