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

doi:10.1090/S0002-9947-2015-06280-1

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

Journal Article•DOI•

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

31 Aug 2015-Transactions of the American Mathematical Society (American Mathematical Society, AMS)-Vol. 367, Iss: 8, pp 5863-5889

TL;DR: In this paper, the exponential growth rate of grid classes of permutations is shown to be equal to the square of the spectral radius of a grid graph, and it is shown that for every γ ≥ 2 + √5 there is a grid class with growth rate arbitrarily close to γ.

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Abstract: Monotone grid classes of permutations have proven very effective in helping to determine structural and enumerative properties of classical permutation pattern classes. 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). Consequently, we utilize spectral graph theoretic results to characterise all slowly growing grid classes and to show that for every γ ≥ 2 + √5 there is a grid class with growth rate arbitrarily close to γ. To prove our main result, we establish bounds on the size of certain families of tours on graphs. In the process, we prove that the family of tours of even length on a connected graph grows at the same rate as the family of "balanced" tours on the graph (in which the number of times an edge is traversed in one direction is the same as the number of times it is traversed in the other direction).

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An Introduction To The Theory Of Graph Spectra

[...]

Sarah Rothstein

01 Jan 2016

TL;DR: An introduction to the theory of graph spectra is available in the book collection an online access to it is set as public so you can download it instantly and is universally compatible with any devices to read.

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Abstract: an introduction to the theory of graph spectra is available in our book collection an online access to it is set as public so you can download it instantly. Our digital library hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the an introduction to the theory of graph spectra is universally compatible with any devices to read.

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222 citations

Some properties of the spectrum of graphs

[...]

ChangAn

01 Jan 1999

TL;DR: In this paper, some properties of the matrix Q(G) are studied and a necessary and sufficient condition for the equality of the spectrum of Q (G) and L (G).

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Abstract: Let G be a graph and denote by Q(G)=D(G)+A(G),L(G)=D(G)-A(G) the sum and the difference between the diagonal matrix of vertex degrees and the adjacency matrix of G,respectively. In this paper,some properties of the matrix Q(G)are studied. At the same time,anecessary and sufficient condition for the equality of the spectrum of Q(G) and L(G) is given.

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125 citations

Posted Content•

Permutations with fixed pattern densities

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Richard Kenyon¹, Daniel Král², Daniel Král³, Charles Radin⁴, Peter Winkler⁵ - Show less +1 more•Institutions (5)

Brown University¹, University of Warwick², Masaryk University³, University of Texas at Austin⁴, Dartmouth College⁵

08 Jun 2015-arXiv: Combinatorics

TL;DR: The limit shapes of random permutations constrained by having fixed densities of a finite number of patterns are shown to be determined by maximizing entropy over permutons with those constraints.

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Abstract: We study scaling limits of random permutations ("permutons") constrained by having fixed densities of a finite number of patterns. We show that the limit shapes are determined by maximizing entropy over permutons with those constraints. In particular, we compute (exactly or numerically) the limit shapes with fixed \hbox{12} density, with fixed \hbox{12} and \hbox{123} densities, with fixed \hbox{12} density and the sum of \hbox{123} and \hbox{213} densities, and with fixed \hbox{123} and \hbox{321} densities. In the last case we explore a particular phase transition. To obtain our results, we also provide a description of permutons using a dynamic construction.

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28 citations

Journal Article•DOI•

On the growth of permutation classes

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David Bevan

01 Jun 2015-arXiv: Combinatorics

TL;DR: It is proved that, asymptotically, patterns in Łukasiewicz paths exhibit a concentrated Gaussian distribution, and a new enumeration technique is introduced, based on associating a graph with each permutation, and the generating functions for some previously unenumerated classes are determined.

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Abstract: We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order. First, we consider monotone grid classes of permutations. We present procedures for calculating the generating function of any class whose matrix has dimensions m × 1 for some m, and of acyclic and unicyclic classes of gridded permutations. We show that almost all large permutations in a grid class have the same shape, and determine this limit shape. 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. In the process, we establish a new result concerning tours on graphs. We also prove a similar result relating the growth rate of a geometric grid class to the matching polynomial of a graph, and determine the effect of edge subdivision on the matching polynomial. We characterise the growth rates of geometric grid classes in terms of the spectral radii of trees. We then investigate the set of growth rates of permutation classes and establish a new upper bound on the value above which every real number is the growth rate of some permutation class. In the process, we prove new results concerning expansions of real numbers in non-integer bases in which the digits are drawn from sets of allowed values. Finally, we introduce a new enumeration technique, based on associating a graph with each permutation, and determine the generating functions for some previously unenumerated classes. We conclude by using this approach to provide an improved lower bound on the growth rate of the class of permutations avoiding the pattern 1324. In the process, we prove that, asymptotically, patterns in Łukasiewicz paths exhibit a concentrated Gaussian distribution.

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22 citations

Journal Article•DOI•

Square permutations are typically rectangular

[...]

Jacopo Borga, Erik Slivken

01 Oct 2020-Annals of Applied Probability

TL;DR: In this article, the authors describe the limit (for two topologies) of large uniform random square permutations, that is, permutations where every point is a record, and consider the limiting behavior of the neighborhood of a point in the permutation through local limits.

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Abstract: We describe the limit (for two topologies) of large uniform random square permutations, that is, permutations where every point is a record. The starting point for all our results is a sampling procedure for asymptotically uniform square permutations. Building on that, we first describe the global behavior by showing that these permutations have a permuton limit which can be described by a random rectangle. We also explore fluctuations about this random rectangle, which we can describe through coupled Brownian motions. Second, we consider the limiting behavior of the neighborhood of a point in the permutation through local limits. As a byproduct, we also determine the random limits of the proportion of occurrences (and consecutive occurrences) of any given pattern in a uniform random square permutation.

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20 citations

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Book•DOI•

Spectra of graphs

[...]

Andries E. Brouwer, Willem H. Haemers

01 Jan 2012

TL;DR: This book gives an elementary treatment of the basic material about graph Spectra, both for ordinary, and Laplace and Seidel spectra, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics.

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Abstract: This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

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2,280 citations

"Growth rates of permutation grid cl..." refers background in this paper

...ernal edge if and only if it contains either a cycle or non-star H graph. An early result of Hoﬀman and Smith [17] shows that the subdivision of an internal edge reduces the spectral radius (also see [11] Proposition 3.1.4 and [13] Theorem 8.1.12). Hence, we can deduce the following unexpected consequence for grid classes: 29 Figure 15: Three unicyclic grid diagrams, of increasing size but decreasing ...
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...he Y graphs (paths with two pendant edges attached to one endvertex) and the three other graphs in Figure 12. For details, see Smith [27] and Lemmens and Seidel [20]; also see [13] Theorem 3.11.1 and [11] Theorem 3.1.3. With these, we can characterise all grid classes with growth rate no greater than 4: 26 Figure 12: A path, a Y graph and the three other connected proper subgraphs of Smith graphs, wit...
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...ize m. A Y graph of size m has spectral radius 2cos π 2m , and the spectral radii of the three other graphs at the right of Figure 12 are 2cos π 12 , 2cos 18 , and 2cos π 30 , from left to right (see [11] 3.1.1). Thus we have the following characterisation of growth rates less than 4: Corollary 4.5. If the growth rate of a monotone grid class is less than 4, it is equal to 4cos2 π k for some k >3. ...
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...classes. So we now present a number of corollaries that follow from spectral graph theoretic considerations. The two recent monographs by Cvetkovi´c, Rowlinson and Simi´c [13] and Brouwer and Haemers [11] provide a valuable overview of spectral graph theory, so, where appropriate, we cite the relevant sections of these (along with the original reference for a result). As a result of Corollary 3.5, cha...
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...s, whose row-column graphs are paths (see the leftmost grid diagram in Figure 12). The spectral radius of a path graph has long been known (Lova´sz and Pelik´an [21]; also see [13] Theorem 8.1.17 and [11] 1.4.4), from which we can conclude: Corollary 4.4. A monotone grid class of size m (having m non-zero cells) whose rowcolumn graph is a path has growth rate 4cos2 π m+2 . This is minimal for any conn...
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Book•

Graphs and homomorphisms

[...]

Pavol Hell, Jaroslav Nešetřil

01 Jan 2004

TL;DR: This chapter discusses the structure of Composition, the partial order of Graphs and Homomorphisms, and testing for the Existence of Homomorphism.

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Abstract: Preface 1. Introduction 2. Products and Retracts 3. The Partial Order of Graphs and Homomorphisms 4. The Structure of Composition 5. Testing for the Existence of Homomorphisms 6. Colouring - Variations on a Theme References Index

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916 citations

Book•

An Introduction to the Theory of Graph Spectra

[...]

Dragoš Cvetković, Peter Rowlinson¹, Slobodan K. Simić•Institutions (1)

University of Stirling¹

01 Oct 2009

TL;DR: In this article, the authors explore the theory of graph spectra, a topic with applications across a wide range of subjects, including computer science, quantum chemistry, electrical engineering and electrical engineering.

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Abstract: This introductory text explores the theory of graph spectra: a topic with applications across a wide range of subjects, including computer science, quantum chemistry and electrical engineering. The spectra examined here are those of the adjacency matrix, the Seidel matrix, the Laplacian, the normalized Laplacian and the signless Laplacian of a finite simple graph. The underlying theme of the book is the relation between the eigenvalues and structure of a graph. Designed as an introductory text for graduate students, or anyone using the theory of graph spectra, this self-contained treatment assumes only a little knowledge of graph theory and linear algebra. The authors include many developments in the field which arise as a result of rapidly expanding interest in the area. Exercises, spectral data and proofs of required results are also provided. The end-of-chapter notes serve as a practical guide to the extensive bibliography of over 500 items.

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730 citations

Journal Article•DOI•

Restricted permutations

[...]

Michael M. Atkinson

28 Jan 1999

TL;DR: The partial order on permutations that underlies the idea of restriction and which gives rise to sets of sequences closed under taking subsequences is studied.

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Abstract: Restricted permutations are those constrained by having to avoid subsequences ordered in various prescribed ways. They have functioned as a convenient descriptor for several sets of permutations which arise naturally in combinatorics and computer science. We study the partial order on permutations that underlies the idea of restriction and which gives rise to sets of sequences closed under taking subsequences. In applications, the question of whether a closed set has a finite basis is often considered. Several constructions that respect the finite basis property are given. A family of closed sets, called profile-closed sets, is introduced and used to solve some instances of the inverse problem:– describing a closed set from its basis. Some enumeration results are also given.

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507 citations

"Growth rates of permutation grid cl..." refers background or methods in this paper

...Atkinson [7] proved that grid classes whose matrices have dimension 1 × m have a finite basis....
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...Moreover, Albert, Atkinson, Bouvel, Ruškuc and Vatter [4] proved a result that implies that if a grid class has an acyclic row-column graph then the generating function of the class is a rational function (the ratio of two polynomials)....
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...Stankova [28] and Kédzy, Snevily and Wang [19] proved that this class is Av(2143, 3412), and Atkinson [6] determined its generating function....
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...In this context the first use of grid classes (but not using that term) was by Atkinson [7], who determined that...
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...More recently, Albert, Atkinson and Brignall [1, 2] and Albert, Atkinson and Vatter [5] have demonstrated the practical uses of grid classes for permutation class enumeration by determining the generating functions of seven permutation classes whose bases consist of two permutations of length four....
[...]

Journal Article•DOI•

Excluded permutation matrices and the Stanley-Wilf conjecture

[...]

Adam Marcus, Gábor Tardos

01 Jul 2004-Journal of Combinatorial Theory, Series A

TL;DR: This paper examines the extremal problem of how many 1-entries an n × n 0-1 matrix can have that avoids a certain fixed submatrix P and proves a linear bound for any permutation matrix P.

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505 citations

"Growth rates of permutation grid cl..." refers background in this paper

...Marcus and Tardos [22] proved the conjecture of Stanley and Wilf that for any permutation class C except the class of all permutations there exists a constant c such that |Ck| 6 ck for all k....
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...Marcus and Tardos [22] proved the conjecture of Stanley and Wilf that for any permutation class C except the class of all permutations there exists a constant c such that |Ck| 6 c for all k....
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