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

2007
*
arXiv
*
pre-print

We consider an infinite

arXiv:0712.1442v1
fatcat:3sbfjdwugbd3dlz6molpjvs6ge
*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. ...##
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On types of growth for graph-different permutations

2009
*
Journal of combinatorial theory. Series A
*

We consider an infinite

doi:10.1016/j.jcta.2008.11.003
fatcat:edtatfzjgjf4lb4iqtbvo4usry
*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." ...##
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On the permutation capacity of digraphs
[article]

2008
*
arXiv
*
pre-print

We extend several results

arXiv:0809.1522v1
fatcat:djrpssavvbg5xckwbriscjx4ka
*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. ...##
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Substitutional subshifts and growth of groups
[article]

2020
*
arXiv
*
pre-print

We show how to use symbolic dynamics

arXiv:2008.04983v1
fatcat:6b7ltcbyirhl7aohjbvbxgrlka
*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. ...##
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Growth rates of permutation classes: categorization up to the uncountability threshold
[article]

2019
*
arXiv
*
pre-print

A significant part

arXiv:1605.04289v3
fatcat:fj5tomw7jfdfdmaxed4iryq7u4
*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. ...##
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On Growth Rates of Permutations, Set Partitions, Ordered Graphs and Other Objects

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*. ...

##
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Large infinite antichains of permutations
[article]

2012
*
arXiv
*
pre-print

We prove the existence and detail the construction

arXiv:1212.3346v2
fatcat:ep5aggpnb5filaugpghfbjazm4
*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 ...##
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Growth rates of permutation classes: from countable to uncountable
[article]

2019
*
arXiv
*
pre-print

We establish that there is an algebraic number ξ≈ 2.30522 such that while there are uncountably many

arXiv:1605.04297v3
fatcat:fbvrvghpzzdepo4geb42yipmmm
*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. ...##
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On growth rates of permutations, set partitions, ordered graphs and other objects
[article]

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 ...

##
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Growing at a Perfect Speed

2009
*
Combinatorics, probability & computing
*

and sal@dcs.st-and.ac.uk A collection

doi:10.1017/s0963548309009699
fatcat:uxa5niy33nf47exhftfec23kuu
*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. ...##
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A graph-theoretic approach to testing associations between disparate sources of functional genomics data

2004
*
Bioinformatics
*

The congruency

doi:10.1093/bioinformatics/bth405
pmid:15256415
fatcat:y6ywvj3hxvbnzlg4z27ggz6hza
*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. ...##
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Oligomorphic Permutation Groups
[chapter]

2009
*
Perspectives in Mathematical Sciences II
*

A

doi:10.1142/9789814273657_0003
fatcat:kxt2dwfkpvc45kfjwjkb2j6qma
*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 . ...##
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Pairwise colliding permutations and the capacity of infinite graphs

2006
*
SIAM Journal on Discrete Mathematics
*

We call two

doi:10.1137/050632877
fatcat:ajl25wgrmrcqdckowsg5cs3cki
*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. ...##
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A Recursive Algorithmic Approach to the Finding of Permutations for the Combination of Any Two Sets
[article]

2014
*
arXiv
*
pre-print

In this paper I present a conjecture

arXiv:1401.1450v1
fatcat:s7u3bmyhvne4lklvsv57bterxu
*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*. ...##
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Economic and Social Institutions: Modelling the Evolution Paths for the Archaic Society

2016
*
Economics & Sociology
*

We use mapping

doi:10.14254/2071-789x.2016/9-2/9
fatcat:vl42ps3asjclfl4j4qvexgnya4
*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). ...
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