Geometric grid classes of permutations

In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and k-letter graphs for a fixed k. However, constructive algorithms are available only for k = 2. In this paper, we present the first constructive polynomial-time algorithm for the recognition of 3-letter graphs. It is based on a structural characterization of graphs in this class. * An extended abstract of this paper was published in [2].

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“…[6], Corollary 1.2.3). This is the language which is in a bijection with Geom(M ), as was shown in [1].…”

Section: Proposition 1 Every Geometric Grid Class Is the Geometric Gmentioning

confidence: 72%

Letter graphs and geometric grid classes of permutations: Characterization and recognition

Alecu

Werra

Zamaraev

2020

Discrete Applied Mathematics

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“…The final part of the present section is devoted to the standard basis of B is strongly rational, meaning that its generating function is rational, together with the generating functions of all of its subclasses [AABRV13]. Here we sketch a description the basis of B k .…”

Section: Proof Sincebmentioning

confidence: 99%

Permutation patterns in genome rearrangement problems: The reversal model

Cerbai

Ferrari

2020

Discrete Applied Mathematics

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In the context of the genome rearrangement problem, we analyze two well known models, namely the reversal and the prefix reversal models, by exploiting the connection with the notion of permutation pattern. More specifically, for any k, we provide a characterization of the set of permutations having distance ≤ k from the identity (which is known to be a permutation class) in terms of what we call generating peg permutations and we describe some properties of its basis, which allow to compute such a basis for small values of k. * Both authors are members of the INdAM Research group GNCS; they are partially supported by INdAM -GNCS 2018 project "Proprietá combinatorie e rilevamento di pattern in strutture discrete lineari e bidimensionali" and by a grant of the "Fondazione della Cassa di Risparmio di Firenze" for the project "Rilevamento di pattern: applicazioni a memorizzazione basata sul DNA, evoluzione del genoma, scelta sociale".

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“…The consideration of monotone grid classes of vectors dates back to the work of Atkinson, Murphy, and Ruškuc and Albert, Atkinson, and Ruškuc , who called them ‘W‐classes’ because the plot of a typical member of the class

Grid (- 1 1 - 1 1)

resembles the letter W. There is a natural encoding of members of monotone grid classes of vectors (later generalized to members of arbitrary geometric grid classes in ).…”

Section: Restricting To a Componentmentioning

confidence: 99%

“…Suppose that

M

is a

0 / \pm 1

matrix of size

t \times 1

(meaning in our notation that it is a row vector of length

t

) and that

C \subseteq Grid (M)

. As we are interested only in growth rates, we consider the set of all

M

‐griddings of members of

scriptC

, denoted by

C^{♯}

, though it is possible to use this encoding to determine the exact enumeration of such classes (see for the vector version and for the more general geometric version). First, label the cells of

M

from 1 to

t

from left‐to‐right (for concreteness only).…”

Section: Restricting To a Componentmentioning

confidence: 99%

“…Geometric grid classes were first introduced by Albert, Atkinson, Bouvel, Ruškuc, and Vatter , who showed that for every

0 / \pm 1

matrix

M

, there is a finite alphabet

normalΣ

(consisting of one letter per nonzero cell of

M

, called the cell alphabet ) and a mapping

φ : {normalΣ}^{*} \to Geom false(M false)

such that whenever

v

is contained in

w

as a subword, then the corresponding permutations satisfy

φ (v) ⩽ φ (w)

. Indeed, this encoding is a generalization of the one used for grid classes of row vectors (which are always geometric grid classes) in the previous section.…”

Section: Well‐quasi‐ordermentioning

confidence: 99%

See 1 more Smart Citation

Growth rates of permutation classes: from countable to uncountable

Vatter

2019

Proc. London Math. Soc.

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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 ξ. Central to the proof are various structural notions regarding generalized grid classes and a new property of permutation classes called concentration. The classification of growth rates up to ξ is completed in a subsequent paper.

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Geometric grid classes of permutations

Cited by 48 publications

References 26 publications

Letter graphs and geometric grid classes of permutations: Characterization and recognition

Letter graphs and geometric grid classes of permutations: Characterization and recognition

Permutation patterns in genome rearrangement problems: The reversal model

Growth rates of permutation classes: from countable to uncountable

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