Growth Rates of Geometric Grid Classes of Permutations

Bevan, David

doi:10.37236/4834

Cited by 11 publications

(19 citation statements)

References 25 publications

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“…Recently, the present author [9] has proved a result similar to Theorem 3.6 for the growth rates of geometric grid classes. Specifically, the growth rate of geometric grid class Geom(M) exists and is equal to the square of the largest root of the matching polynomial of the row-column graph of what is known as the "double refinement" of matrix M. This value coincides with ρ(G(M)) 2 for acyclic classes, Geom(M) and Grid(M) being identical when G(M) is a forest.…”

Section: Discussionmentioning

confidence: 52%

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

Bevan

2015

Trans. Amer. Math. Soc.

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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). 5863 Licensed to Univ of Mississippi. Prepared on Tue Jul 14 02:56:52 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5864 DAVID BEVANClearly, the containment relation is a partial order on the set of all permutations. A classical permutation class (or "pattern class") is a set of permutations closed downwards (a down-set) in this partial order. From a graphical perspective, this means that erasing points from the plot of a permutation in a permutation class C always results in the plot of another permutation in C when the axes are rescaled appropriately.Given a permutation class C, we denote by C k = {σ ∈ C : |σ| = k} the set of permutations in C of length k. The (ordinary) generating function of C is thus k∈N |C k |z k = σ∈C z |σ| . It is common to define a permutation class C "negatively" by stating the minimal set of permutations B that do not occur in the class. In this case, we write C = Av(B) (where Av signifies "avoids"). B is called the basis of C. The basis of a permutation class is an antichain (a set of pairwise incomparable elements) and may be infinite. Licensed to Univ of Mississippi. Prepared on Tue Jul 14 02:56:52 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GROWTH RATES OF PERMUTATION GRID CLASSES 5865 basis has finite lower and upper exponential growth rates defined, respectively, by gr(C) = lim inf k→∞ |C k | 1/k and gr(C) = lim sup k→∞ |C k | 1/k .If the lower and upper growth rates coincide, then C has a growth rate, which we denote gr(C). (It is widely conjectured that every permutation class has a growth rate.) In [30], Vatter investigated the possible values of permutation class growth rates, and used generalised grid classes to characterize all the (countably many) permutation classes with growth rates below κ ≈ 2.20557. He also established that there are uncountably many permutation classes with growth rate κ, and in a separate...

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Section: Discussionmentioning

confidence: 52%

“…Using results concerning graphs with small spectral radius, we can characterise grid classes with growth rates no greater than 9 2 . This is similar to Vatter's characterisation of "small" permutation classes (with growth rate less than κ ≈ 2.20557) in [30].…”

Section: Slowly Growing Grid Classesmentioning

confidence: 99%

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

Bevan

2015

Trans. Amer. Math. Soc.

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“…We believe that the techniques introduced in this work-especially the panel encoding of Section 5will find many more applications. To introduce these we first observe that in the language of geometric grid classes [2,8,14], the 321-avoiding permutations form the grid class of the infinite matrix …”

Section: Resultsmentioning

confidence: 99%

Rationality for subclasses of 321-avoiding permutations

Albert

Brignall

Ruškuc

et al. 2019

European Journal of Combinatorics

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We prove that every proper subclass of the 321-avoiding permutations that is defined either by only finitely many additional restrictions or is well-quasi-ordered has a rational generating function. To do so we show that any such class is in bijective correspondence with a regular language. The proof makes significant use of formal languages and of a host of encodings, including a new mapping called the panel encoding that maps languages over the infinite alphabet of positive integers avoiding certain subwords to languages over finite alphabets.

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“…and can be drawn on a series of line segments each with slope ±1, as determined by σ. Any π ∈ S σ n := S σ ∩ S n must admit a σ-segmentation defined in [4] (and referred to as a gridding in [1,8,9]) to be a sequence e = (e 0 , e 1 , . .…”

Section: The σ-Classesmentioning

confidence: 99%

Enumeration of cyclic permutations in vector grid classes

Archer

Lauderdale

2020

Journal of Combinatorics

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A grid class consists of permutations whose pictorial depiction can be partitioned into increasing and decreasing parts as determined by a given matrix. In this paper, we introduce a method for enumerating cyclic permutations in vector grid classes by establishing a bijective relationship with certain necklaces. We use this method to complete the enumeration of cyclic permutations in the length 3 vector grid classes. In addition, we define an analog of Wilf-equivalence between these sets. We conclude by discussing cyclic permutations in alternating grid classes.

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Growth Rates of Geometric Grid Classes of Permutations

Cited by 11 publications

References 25 publications

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

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

Rationality for subclasses of 321-avoiding permutations

Enumeration of cyclic permutations in vector grid classes

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