Substitution-closed pattern classes

Atkinson, M. D.; Ruškuc, Nik; Smith, Rebecca

doi:10.1016/j.jcta.2010.10.006

Cited by 11 publications

(14 citation statements)

References 14 publications

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“…The simple permutations form the elemental building blocks upon which all other permutations are constructed by means of the substitution decomposition. 3 Analogues of this decomposition exist for every relational structure, and it has frequently arisen in a wide variety of perspectives, ranging from game theory to combinatorial optimization -for references see Möhring [35] or Möhring and Radermacher [36]. Its first appearance seems to be in a 1953 talk by Fraïssé (though only the abstract of this talk [25] survives).…”

Section: Substitution Decompositionmentioning

confidence: 99%

A survey of simple permutations

Brignall

2010

Permutation Patterns

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We survey the known results about simple permutations. In particular, we present a number of recent enumerative and structural results pertaining to simple permutations, and show how simple permutations play an important role in the study of permutation classes. We demonstrate how classes containing only finitely many simple permutations satisfy a number of special properties relating to enumeration, partial well-order and the property of being finitely based. 2 In the past, permutation classes have also been called closed classes or pattern classes.

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Section: Substitution Decompositionmentioning

confidence: 99%

A survey of simple permutations

Brignall

2010

Permutation Patterns

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“…

\begin{matrix} left Inflation & left Sequence of sum & left Sum closure & left Sequence of skew & left Skew closure \\ left of 25314 & left indecomposables & left growth rate & left indecomposables & left growth rate \\ left 256314 & left 1, 1, 3, 6, 4, 1 & left \approx 2.44874 & left 1, 1, 3, 4, 3, 1 & left \approx 2.36772 \\ left 265314 & left 1, 1, 3, 5, 4, 1 & left \approx 2.41421 & left 1, 1, 3, 4, 3, 1 & left \approx 2.36772 \\ left 263415 & left 1, 1, 3, 6, 3, 1 & left \approx 2.43245 & left 1, 1, 3, 4, 3, 1 & left \approx 2.36772 \\ left 264315 & left 1, 1, 3, 4, 3, 1 & left \approx 2.36772 & left 1, 1, 3, 6, 3, 1 & left \approx 2.43245 \end{matrix}

Next, we show that the simple permutations of

G_{2.36}

are contained in

Av (321) \cup Av (123) \cup {25314, 41352}

. For this, we use a result of Atkinson, Ruškuc, and Smith . They showed that the substitution closure of a principally based class is typically infinitely based.…”

Section: Gridding Intermediate Classesmentioning

confidence: 99%

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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“…We may then choose an element x 2 ∈ X to be minimal such that x 1 , x 2 begins a bad sequence. Proceeding by induction 17 , if we assume that x 1 , x 2 , . .…”

Section: Proposition 52 (Nash-williams [79 Proof Of Lemma 2]) a Well-...mentioning

confidence: 99%

“…20 See Brignall and Vatter [30] for a proof of Schmerl and Trotter's theorem in the special case of permutations. 21 However, in practice it can be difficult to establish precisely what the members of the basis are, and there are frequently infinitely many of them-see Atkinson, Ruškuc, and Smith [17] for one such example.…”

Section: Proposition 710 (Albert and Atkinson [1]) The Basis Of The S...mentioning

confidence: 99%

Labelled well-quasi-order for permutation classes

Brignall¹,

Vatter²

2021

Preprint

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While the theory of labeled well-quasi-order has received significant attention in the graph setting, it has not yet been considered in the context of permutation patterns. We initiate this study here, and using labeled well-quasi-order are able to subsume and extend all of the well-quasiorder results in the permutation patterns literature. Connections to the graph setting are emphasized throughout. In particular, we establish that a permutation class is labeled well-quasi-ordered if and only if its corresponding graph class is also labeled well-quasi-ordered. * Vatter's research was partially supported by the Simons Foundation via award number 636113. 1 Our figure of 80 years dates the study of well-quasi-order to Wagner [103]. 2 A well-quasi-ordered partial order is sometimes called a partially well-ordered or well-partially-ordered set, or it is simply called a partial well-order. In particular, these terms are used in some of the early work on well-quasi-order in the permutation patterns context. We tend to agree with Kruskal's sentiment from [63, p. 298], where he wrote that "at the casual level it is easier to work with [partial orders] than [quasi-orders], but in advanced work the reverse is true.

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Substitution-closed pattern classes

Cited by 11 publications

References 14 publications

A survey of simple permutations

A survey of simple permutations

Growth rates of permutation classes: from countable to uncountable

Labelled well-quasi-order for permutation classes

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