Wreath-closed permutation classes Unique Embedding Finite types - PowerPoint PPT Presentation

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Introduction Pin Sequences Wreath-closed permutation classes Unique Embedding Finite types Indecomposable permutations of finite type Decomposable permutations Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith of finite type Spiral permutations Infinite types Decomposable permutations of infinite type Indecomposable permutations of infinite type June 17, 2008 Summary

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Introduction Pin Sequences Definition Unique Embedding The basis of a pattern class is the maximum set of Finite types minimal permutations that are avoided by all Indecomposable permutations of finite type permutations in the pattern class. Decomposable permutations of finite type Spiral permutations Infinite types Note that the set of permutations avoiding a particular Decomposable permutations of infinite type permutation or set of permutations is a closed class. Indecomposable permutations of infinite type Summary

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith The permutation α [ β 1 , . . . , β n ] is such that the i th term Introduction of α is substituted by β i . In other words α [ β 1 , . . . , β n ] Pin Sequences consists of n segments order isomorphic to β 1 , . . . , β n Unique Embedding where the relative order of the segments is the same as Finite types Indecomposable the relative order of the terms of α . permutations of finite type Decomposable permutations of finite type Spiral permutations Infinite types Decomposable permutations of infinite type Indecomposable permutations of infinite type Summary

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith The permutation α [ β 1 , . . . , β n ] is such that the i th term Introduction of α is substituted by β i . In other words α [ β 1 , . . . , β n ] Pin Sequences consists of n segments order isomorphic to β 1 , . . . , β n Unique Embedding where the relative order of the segments is the same as Finite types Indecomposable the relative order of the terms of α . permutations of finite type Decomposable permutations of finite type Spiral permutations Infinite types Example Decomposable permutations of infinite type Let α = 2413 , β 1 = 123 , β 2 = 21 , β 3 = 1 , β = 312. Then Indecomposable permutations of infinite type α [ β 1 , . . . , β 4 ] = 234 98 1 756. Summary

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Introduction Pin Sequences Unique Embedding Finite types Indecomposable permutations of finite type Decomposable permutations of finite type Spiral permutations Infinite types Decomposable permutations of infinite type Indecomposable permutations of infinite type Summary Figure: α [ β 1 , . . . , β 4 ] = 234 98 1 756.

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Definition Introduction A pattern class X is said to be wreath-closed (or Pin Sequences substitution closed) if α [ β 1 , . . . , β n ] ∈ X for all Unique Embedding α, β 1 , . . . , β n ∈ X . Finite types Indecomposable permutations of finite type Decomposable permutations of finite type Spiral permutations Infinite types Decomposable permutations of infinite type Indecomposable permutations of infinite type Summary

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Definition Introduction A pattern class X is said to be wreath-closed (or Pin Sequences substitution closed) if α [ β 1 , . . . , β n ] ∈ X for all Unique Embedding α, β 1 , . . . , β n ∈ X . Finite types Indecomposable permutations of finite type Decomposable permutations The intersection of wreath-closed pattern classes is itself of finite type Spiral permutations wreath-closed. Thus any pattern class is contained in a Infinite types smallest wreath-closed class which is referred to as its Decomposable permutations of infinite type wreath closure . Indecomposable permutations of infinite type Summary

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Definition Introduction A pattern class X is said to be wreath-closed (or Pin Sequences substitution closed) if α [ β 1 , . . . , β n ] ∈ X for all Unique Embedding α, β 1 , . . . , β n ∈ X . Finite types Indecomposable permutations of finite type Decomposable permutations The intersection of wreath-closed pattern classes is itself of finite type Spiral permutations wreath-closed. Thus any pattern class is contained in a Infinite types smallest wreath-closed class which is referred to as its Decomposable permutations of infinite type wreath closure . Indecomposable permutations of infinite type In this talk, we consider the wreath-closure of X = Av ( ψ ) Summary where ψ is any permutation. In particular, does the wreath-closure of X have a finite or infinite basis?

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Definition An interval of a permutation is a consecutive sequence of Introduction elements of the permutation that are also form a set of Pin Sequences consecutive (integer) values. Unique Embedding Finite types Indecomposable permutations of finite type Decomposable permutations of finite type Spiral permutations Infinite types Decomposable permutations of infinite type Indecomposable permutations of infinite type Summary

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Definition An interval of a permutation is a consecutive sequence of Introduction elements of the permutation that are also form a set of Pin Sequences consecutive (integer) values. Unique Embedding Finite types Indecomposable permutations of finite type Decomposable permutations Definition of finite type Spiral permutations A simple permutation is a permutation whose only Infinite types intervals are singletons and the entire permutation. Decomposable permutations of infinite type Indecomposable permutations of infinite type Summary

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Definition An interval of a permutation is a consecutive sequence of Introduction elements of the permutation that are also form a set of Pin Sequences consecutive (integer) values. Unique Embedding Finite types Indecomposable permutations of finite type Decomposable permutations Definition of finite type Spiral permutations A simple permutation is a permutation whose only Infinite types intervals are singletons and the entire permutation. Decomposable permutations of infinite type Indecomposable permutations of infinite type Summary Example The permutation 58147362 is simple.

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Introduction Pin Sequences Unique Embedding Finite types Indecomposable permutations of finite type Decomposable permutations of finite type Spiral permutations Infinite types Decomposable permutations of infinite type Indecomposable permutations of infinite type Summary Figure: The permutation 58147362.

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Introduction In their article “Simple permutations and pattern Pin Sequences restricted permutations”, Albert and Atkinson show the Unique Embedding crucial connection between wreath-closed classes and Finite types Indecomposable simple permutations: permutations of finite type Decomposable permutations of finite type Proposition Spiral permutations Infinite types A pattern class is wreath-closed if and only if its basis Decomposable permutations of infinite type consists of simple permutations. Indecomposable permutations of infinite type Summary

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Definition Introduction An extension ξ of ψ is a minimal simple extension of ψ if Pin Sequences Unique Embedding 1. ξ is simple, and Finite types 2. among all simple extensions of ψ , ξ is minimal under Indecomposable permutations of finite type the subpermutation order. Decomposable permutations of finite type Spiral permutations Infinite types Decomposable permutations of infinite type Indecomposable permutations of infinite type Summary

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Definition Introduction An extension ξ of ψ is a minimal simple extension of ψ if Pin Sequences Unique Embedding 1. ξ is simple, and Finite types 2. among all simple extensions of ψ , ξ is minimal under Indecomposable permutations of finite type the subpermutation order. Decomposable permutations of finite type Spiral permutations Infinite types Decomposable permutations of infinite type Indecomposable Lemma permutations of infinite type The basis of the wreath closure of X = Av ( ψ ) is the set Summary of minimal simple extensions of ψ .

Wreath-closed permutation classes Mike Atkinson, Nik Ruˇ skuc, and Rebecca Smith Example Introduction Pin Sequences A case where the wreath-closure of X = Av ( ψ ) has an Unique Embedding infinite basis. Finite types Indecomposable permutations of finite type Decomposable permutations of finite type Spiral permutations Infinite types Decomposable permutations of infinite type Indecomposable permutations of infinite type Summary