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Simple Permutations
R.L.F. Brignalljoint work with Sophie Huczynska, Nik Ruškuc and Vincent Vatter
School of Mathematics and StatisticsUniversity of St Andrews
Thursday 15th June, 2006
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Introduction
1 Basic ConceptsPermutation ClassesIntervals and Simple Permutations
2 Algebraic Generating Functions for Sets of PermutationsFinitely Many SimplesSets of Permutations
3 A Decomposition Theorem with Enumerative ConsequencesAimPin SequencesDecomposing Simple Permutations
4 Decidability and Unavoidable StructuresMore on PinsDecidability
![Page 3: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/3.jpg)
Basic Concepts
Outline
1 Basic ConceptsPermutation ClassesIntervals and Simple Permutations
2 Algebraic Generating Functions for Sets of PermutationsFinitely Many SimplesSets of Permutations
3 A Decomposition Theorem with Enumerative ConsequencesAimPin SequencesDecomposing Simple Permutations
4 Decidability and Unavoidable StructuresMore on PinsDecidability
![Page 4: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/4.jpg)
Basic Concepts
Permutation Classes
Pattern Involvement
Regard a permutation of length n as an ordering of thesymbols 1, . . . , n.
A permutation τ = t1t2 . . . tk is involved in the permutationσ = s1s2 . . . sn if there exists a subsequence si1 , si2 , . . . , sikorder isomorphic to τ .
Example
![Page 5: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/5.jpg)
Basic Concepts
Permutation Classes
Pattern Involvement
Regard a permutation of length n as an ordering of thesymbols 1, . . . , n.
A permutation τ = t1t2 . . . tk is involved in the permutationσ = s1s2 . . . sn if there exists a subsequence si1 , si2 , . . . , sikorder isomorphic to τ .
Example
![Page 6: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/6.jpg)
Basic Concepts
Permutation Classes
Pattern Involvement
Regard a permutation of length n as an ordering of thesymbols 1, . . . , n.
A permutation τ = t1t2 . . . tk is involved in the permutationσ = s1s2 . . . sn if there exists a subsequence si1 , si2 , . . . , sikorder isomorphic to τ .
Example
1 3 5 2 4 4 2 1 6 3 8 5 7
![Page 7: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/7.jpg)
Basic Concepts
Permutation Classes
Pattern Involvement
Regard a permutation of length n as an ordering of thesymbols 1, . . . , n.
A permutation τ = t1t2 . . . tk is involved in the permutationσ = s1s2 . . . sn if there exists a subsequence si1 , si2 , . . . , sikorder isomorphic to τ .
Example
1 3 5 2 4 ≺ 4 2 1 6 3 8 5 7
![Page 8: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/8.jpg)
Basic Concepts
Permutation Classes
Permutation Classes
Involvement forms a partial order on the set of allpermutations.
Downsets of permutations in this partial order formpermutation classes.
A permutation class C can be seen to avoid certainpermutations. Write C = Av(B).
Example
The class C = Av(12) consists of all the decreasingpermutations:
{1, 21, 321, 4321, . . .}
![Page 9: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/9.jpg)
Basic Concepts
Permutation Classes
Permutation Classes
Involvement forms a partial order on the set of allpermutations.
Downsets of permutations in this partial order formpermutation classes.
A permutation class C can be seen to avoid certainpermutations. Write C = Av(B).
Example
The class C = Av(12) consists of all the decreasingpermutations:
{1, 21, 321, 4321, . . .}
![Page 10: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/10.jpg)
Basic Concepts
Permutation Classes
Permutation Classes
Involvement forms a partial order on the set of allpermutations.
Downsets of permutations in this partial order formpermutation classes.
A permutation class C can be seen to avoid certainpermutations. Write C = Av(B).
Example
The class C = Av(12) consists of all the decreasingpermutations:
{1, 21, 321, 4321, . . .}
![Page 11: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/11.jpg)
Basic Concepts
Permutation Classes
Permutation Classes
Involvement forms a partial order on the set of allpermutations.
Downsets of permutations in this partial order formpermutation classes.
A permutation class C can be seen to avoid certainpermutations. Write C = Av(B).
Example
The class C = Av(12) consists of all the decreasingpermutations:
{1, 21, 321, 4321, . . .}
![Page 12: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/12.jpg)
Basic Concepts
Permutation Classes
Permutation Classes
Involvement forms a partial order on the set of allpermutations.
Downsets of permutations in this partial order formpermutation classes.
A permutation class C can be seen to avoid certainpermutations. Write C = Av(B).
Example
The class C = Av(12) consists of all the decreasingpermutations:
{1, 21, 321, 4321, . . .}
![Page 13: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/13.jpg)
Basic Concepts
Permutation Classes
Generating Functions
Cn – permutations in C of length n.∑|Cn|xn is the generating function.
Example
The generating function of C = Av(12) is:
1 + x + x2 + x3 + · · · =1
1 − x
![Page 14: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/14.jpg)
Basic Concepts
Permutation Classes
Generating Functions
Cn – permutations in C of length n.∑|Cn|xn is the generating function.
Example
The generating function of C = Av(12) is:
1 + x + x2 + x3 + · · · =1
1 − x
![Page 15: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/15.jpg)
Basic Concepts
Permutation Classes
Generating Functions
Cn – permutations in C of length n.∑|Cn|xn is the generating function.
Example
The generating function of C = Av(12) is:
1 + x + x2 + x3 + · · · =1
1 − x
![Page 16: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/16.jpg)
Basic Concepts
Permutation Classes
Generating Functions
Cn – permutations in C of length n.∑|Cn|xn is the generating function.
Example
The generating function of C = Av(12) is:
1 + x + x2 + x3 + · · · =1
1 − x
![Page 17: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/17.jpg)
Basic Concepts
Intervals and Simple Permutations
Intervals
Pick any permutation π.
An interval of π is a set of contiguous indices I = [a, b]such that π(I) = {π(i) : i ∈ I} is also contiguous.
Example
![Page 18: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/18.jpg)
Basic Concepts
Intervals and Simple Permutations
Intervals
Pick any permutation π.
An interval of π is a set of contiguous indices I = [a, b]such that π(I) = {π(i) : i ∈ I} is also contiguous.
Example
![Page 19: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/19.jpg)
Basic Concepts
Intervals and Simple Permutations
Intervals
Pick any permutation π.
An interval of π is a set of contiguous indices I = [a, b]such that π(I) = {π(i) : i ∈ I} is also contiguous.
Example
![Page 20: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/20.jpg)
Basic Concepts
Intervals and Simple Permutations
Intervals
Pick any permutation π.
An interval of π is a set of contiguous indices I = [a, b]such that π(I) = {π(i) : i ∈ I} is also contiguous.
Example
![Page 21: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/21.jpg)
Basic Concepts
Intervals and Simple Permutations
Simple Permutations
Only intervals are singletons and the whole thing.
Example
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Basic Concepts
Intervals and Simple Permutations
Simple Permutations
Only intervals are singletons and the whole thing.
Example
![Page 23: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/23.jpg)
Basic Concepts
Intervals and Simple Permutations
Simple Permutations
Only intervals are singletons and the whole thing.
Example
![Page 24: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/24.jpg)
Basic Concepts
Intervals and Simple Permutations
Simple Permutations
Only intervals are singletons and the whole thing.
Example
![Page 25: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/25.jpg)
Basic Concepts
Intervals and Simple Permutations
Simple Permutations
Only intervals are singletons and the whole thing.
Example
![Page 26: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/26.jpg)
Basic Concepts
Intervals and Simple Permutations
Simple Permutations
Only intervals are singletons and the whole thing.
Example
![Page 27: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/27.jpg)
Basic Concepts
Intervals and Simple Permutations
Simple Permutations
Only intervals are singletons and the whole thing.
Example
![Page 28: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/28.jpg)
Basic Concepts
Intervals and Simple Permutations
Special Simple Permutations
Parallel alternations.
Wedge permutations
Two flavours of wedge simple permutation.
![Page 29: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/29.jpg)
Basic Concepts
Intervals and Simple Permutations
Special Simple Permutations
Parallel alternations.
Wedge permutations
Two flavours of wedge simple permutation.
![Page 30: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/30.jpg)
Basic Concepts
Intervals and Simple Permutations
Special Simple Permutations
Parallel alternations.
Wedge permutations – not simple!
Two flavours of wedge simple permutation.
![Page 31: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/31.jpg)
Basic Concepts
Intervals and Simple Permutations
Special Simple Permutations
Parallel alternations.
Wedge permutations
Two flavours of wedge simple permutation.
![Page 32: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/32.jpg)
Algebraic Generating Functions for Sets of Permutations
Outline
1 Basic ConceptsPermutation ClassesIntervals and Simple Permutations
2 Algebraic Generating Functions for Sets of PermutationsFinitely Many SimplesSets of Permutations
3 A Decomposition Theorem with Enumerative ConsequencesAimPin SequencesDecomposing Simple Permutations
4 Decidability and Unavoidable StructuresMore on PinsDecidability
![Page 33: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/33.jpg)
Algebraic Generating Functions for Sets of Permutations
Finitely Many Simples
Motivation
Example
Av(132)
Av(132)
132-avoiders: generic structure.
Only simple permutations are1, 12, and 21.
Enumerate recursively: f (x) = xf (x)2 + 1.
![Page 34: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/34.jpg)
Algebraic Generating Functions for Sets of Permutations
Finitely Many Simples
Motivation
Example
Av(132)
Av(132)
132-avoiders: generic structure.
Only simple permutations are1, 12, and 21.
Enumerate recursively: f (x) = xf (x)2 + 1.
![Page 35: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/35.jpg)
Algebraic Generating Functions for Sets of Permutations
Finitely Many Simples
Motivation
Example
Av(132)
Av(132)
132-avoiders: generic structure.
Only simple permutations are1, 12, and 21.
Enumerate recursively: f (x) = xf (x)2 + 1.
![Page 36: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/36.jpg)
Algebraic Generating Functions for Sets of Permutations
Finitely Many Simples
Motivation II
“ ...the standard intuition of what a family with an algebraicgenerating function looks like: the algebraicity suggeststhat it may (or should...), be possible to give a recursivedescription of the objects based on disjoint union of setsand concatentation of objects.”
— Bousquet-Mélou, 2006
We can always write permutations with a simple blockpattern, the substitution decomposition.
Use recursive enumeration for classes with finitely manysimple permutations.
Expect an algebraic generating function.
![Page 37: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/37.jpg)
Algebraic Generating Functions for Sets of Permutations
Finitely Many Simples
Motivation II
“ ...the standard intuition of what a family with an algebraicgenerating function looks like: the algebraicity suggeststhat it may (or should...), be possible to give a recursivedescription of the objects based on disjoint union of setsand concatentation of objects.”
— Bousquet-Mélou, 2006
We can always write permutations with a simple blockpattern, the substitution decomposition.
Use recursive enumeration for classes with finitely manysimple permutations.
Expect an algebraic generating function.
![Page 38: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/38.jpg)
Algebraic Generating Functions for Sets of Permutations
Finitely Many Simples
Motivation II
“ ...the standard intuition of what a family with an algebraicgenerating function looks like: the algebraicity suggeststhat it may (or should...), be possible to give a recursivedescription of the objects based on disjoint union of setsand concatentation of objects.”
— Bousquet-Mélou, 2006
We can always write permutations with a simple blockpattern, the substitution decomposition.
Use recursive enumeration for classes with finitely manysimple permutations.
Expect an algebraic generating function.
![Page 39: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/39.jpg)
Algebraic Generating Functions for Sets of Permutations
Finitely Many Simples
Motivation II
“ ...the standard intuition of what a family with an algebraicgenerating function looks like: the algebraicity suggeststhat it may (or should...), be possible to give a recursivedescription of the objects based on disjoint union of setsand concatentation of objects.”
— Bousquet-Mélou, 2006
We can always write permutations with a simple blockpattern, the substitution decomposition.
Use recursive enumeration for classes with finitely manysimple permutations.
Expect an algebraic generating function.
![Page 40: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/40.jpg)
Algebraic Generating Functions for Sets of Permutations
Sets of Permutations
Algebraic Generating Functions
Theorem (RB, SH, VV)
In a permutation class C with only finitely many simplepermutations, the following sequences have algebraicgenerating functions:
the number of permutations in Cn (Albert and Atkinson),
the number of alternating permutations in Cn,
the number of even permutations in Cn,
the number of Dumont permutations in Cn,
the number of permutations in Cn avoiding any finite set ofblocked or barred permutations,
the number of involutions in Cn, and
Any (finite) combination of the above.
![Page 41: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/41.jpg)
Algebraic Generating Functions for Sets of Permutations
Sets of Permutations
Algebraic Generating Functions
Theorem (RB, SH, VV)
In a permutation class C with only finitely many simplepermutations, the following sequences have algebraicgenerating functions:
the number of permutations in Cn (Albert and Atkinson),
the number of alternating permutations in Cn,
the number of even permutations in Cn,
the number of Dumont permutations in Cn,
the number of permutations in Cn avoiding any finite set ofblocked or barred permutations,
the number of involutions in Cn, and
Any (finite) combination of the above.
![Page 42: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/42.jpg)
Algebraic Generating Functions for Sets of Permutations
Sets of Permutations
Algebraic Generating Functions
Theorem (RB, SH, VV)
In a permutation class C with only finitely many simplepermutations, the following sequences have algebraicgenerating functions:
the number of permutations in Cn (Albert and Atkinson),
the number of alternating permutations in Cn,
the number of even permutations in Cn,
the number of Dumont permutations in Cn,
the number of permutations in Cn avoiding any finite set ofblocked or barred permutations,
the number of involutions in Cn, and
Any (finite) combination of the above.
![Page 43: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/43.jpg)
Algebraic Generating Functions for Sets of Permutations
Sets of Permutations
Algebraic Generating Functions
Theorem (RB, SH, VV)
In a permutation class C with only finitely many simplepermutations, the following sequences have algebraicgenerating functions:
the number of permutations in Cn (Albert and Atkinson),
the number of alternating permutations in Cn,
the number of even permutations in Cn,
the number of Dumont permutations in Cn,
the number of permutations in Cn avoiding any finite set ofblocked or barred permutations,
the number of involutions in Cn, and
Any (finite) combination of the above.
![Page 44: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/44.jpg)
Algebraic Generating Functions for Sets of Permutations
Sets of Permutations
Algebraic Generating Functions
Theorem (RB, SH, VV)
In a permutation class C with only finitely many simplepermutations, the following sequences have algebraicgenerating functions:
the number of permutations in Cn (Albert and Atkinson),
the number of alternating permutations in Cn,
the number of even permutations in Cn,
the number of Dumont permutations in Cn,
the number of permutations in Cn avoiding any finite set ofblocked or barred permutations,
the number of involutions in Cn, and
Any (finite) combination of the above.
![Page 45: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/45.jpg)
Algebraic Generating Functions for Sets of Permutations
Sets of Permutations
Algebraic Generating Functions
Theorem (RB, SH, VV)
In a permutation class C with only finitely many simplepermutations, the following sequences have algebraicgenerating functions:
the number of permutations in Cn (Albert and Atkinson),
the number of alternating permutations in Cn,
the number of even permutations in Cn,
the number of Dumont permutations in Cn,
the number of permutations in Cn avoiding any finite set ofblocked or barred permutations,
the number of involutions in Cn, and
Any (finite) combination of the above.
![Page 46: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/46.jpg)
Algebraic Generating Functions for Sets of Permutations
Sets of Permutations
Algebraic Generating Functions
Theorem (RB, SH, VV)
In a permutation class C with only finitely many simplepermutations, the following sequences have algebraicgenerating functions:
the number of permutations in Cn (Albert and Atkinson),
the number of alternating permutations in Cn,
the number of even permutations in Cn,
the number of Dumont permutations in Cn,
the number of permutations in Cn avoiding any finite set ofblocked or barred permutations,
the number of involutions in Cn, and
Any (finite) combination of the above.
![Page 47: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/47.jpg)
A Decomposition Theorem with Enumerative Consequences
Outline
1 Basic ConceptsPermutation ClassesIntervals and Simple Permutations
2 Algebraic Generating Functions for Sets of PermutationsFinitely Many SimplesSets of Permutations
3 A Decomposition Theorem with Enumerative ConsequencesAimPin SequencesDecomposing Simple Permutations
4 Decidability and Unavoidable StructuresMore on PinsDecidability
![Page 48: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/48.jpg)
A Decomposition Theorem with Enumerative Consequences
Aim
What we want to find
Large simple permutation, size f (k).
Find two simple permutations inside, each of size k .
Overlap of at most two points – almost disjoint.
![Page 49: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/49.jpg)
A Decomposition Theorem with Enumerative Consequences
Aim
What we want to find
Large simple permutation, size f (k).
Find two simple permutations inside, each of size k .
Overlap of at most two points – almost disjoint.
![Page 50: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/50.jpg)
A Decomposition Theorem with Enumerative Consequences
Aim
What we want to find
Large simple permutation, size f (k).
Find two simple permutations inside, each of size k .
Overlap of at most two points – almost disjoint.
![Page 51: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/51.jpg)
A Decomposition Theorem with Enumerative Consequences
Aim
What we want to find
Large simple permutation, size f (k).
Find two simple permutations inside, each of size k .
Overlap of at most two points – almost disjoint.
![Page 52: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/52.jpg)
A Decomposition Theorem with Enumerative Consequences
Aim
Why I
Every simple of length ≥ 4 contains 132.
Every simple of length ≥ f (4) contains 2 almost disjointcopies of 132.
≥ f (f (4)) contains 4 copies of 132.
...
Theorem (Bóna; Mansour and Vainshtein)
For every fixed r, the class of all permutations containing atmost r copies of 132 has an algebraic generating function.
![Page 53: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/53.jpg)
A Decomposition Theorem with Enumerative Consequences
Aim
Why I
Every simple of length ≥ 4 contains 132.
Every simple of length ≥ f (4) contains 2 almost disjointcopies of 132.
≥ f (f (4)) contains 4 copies of 132.
...
Theorem (Bóna; Mansour and Vainshtein)
For every fixed r, the class of all permutations containing atmost r copies of 132 has an algebraic generating function.
![Page 54: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/54.jpg)
A Decomposition Theorem with Enumerative Consequences
Aim
Why I
Every simple of length ≥ 4 contains 132.
Every simple of length ≥ f (4) contains 2 almost disjointcopies of 132.
≥ f (f (4)) contains 4 copies of 132.
...
Theorem (Bóna; Mansour and Vainshtein)
For every fixed r, the class of all permutations containing atmost r copies of 132 has an algebraic generating function.
![Page 55: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/55.jpg)
A Decomposition Theorem with Enumerative Consequences
Aim
Why I
Every simple of length ≥ 4 contains 132.
Every simple of length ≥ f (4) contains 2 almost disjointcopies of 132.
≥ f (f (4)) contains 4 copies of 132.
...
Theorem (Bóna; Mansour and Vainshtein)
For every fixed r, the class of all permutations containing atmost r copies of 132 has an algebraic generating function.
![Page 56: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/56.jpg)
A Decomposition Theorem with Enumerative Consequences
Aim
Why II
Av(β≤r11 , β
≤r22 , . . . , β
≤rkk ) — the class with: ≤ r1 copies of β1,
≤ r2 copies of β2, etc.
Corollary
If the class Av(β1, β2, . . . , βk ) contains only finitely many simplepermutations then for all choices of nonnegative integers r1, r2,. . . , and rk , the class Av(β≤r1
1 , β≤r22 , . . . , β
≤rkk ) also contains only
finitely many simple permutations.
Corollary
For all r and s, every subclass of Av(2413≤r , 3142≤s) containsonly finitely many simple permutations and thus has analgebraic generating function.
![Page 57: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/57.jpg)
A Decomposition Theorem with Enumerative Consequences
Aim
Why II
Av(β≤r11 , β
≤r22 , . . . , β
≤rkk ) — the class with: ≤ r1 copies of β1,
≤ r2 copies of β2, etc.
Corollary
If the class Av(β1, β2, . . . , βk ) contains only finitely many simplepermutations then for all choices of nonnegative integers r1, r2,. . . , and rk , the class Av(β≤r1
1 , β≤r22 , . . . , β
≤rkk ) also contains only
finitely many simple permutations.
Corollary
For all r and s, every subclass of Av(2413≤r , 3142≤s) containsonly finitely many simple permutations and thus has analgebraic generating function.
![Page 58: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/58.jpg)
A Decomposition Theorem with Enumerative Consequences
Aim
Why II
Av(β≤r11 , β
≤r22 , . . . , β
≤rkk ) — the class with: ≤ r1 copies of β1,
≤ r2 copies of β2, etc.
Corollary
If the class Av(β1, β2, . . . , βk ) contains only finitely many simplepermutations then for all choices of nonnegative integers r1, r2,. . . , and rk , the class Av(β≤r1
1 , β≤r22 , . . . , β
≤rkk ) also contains only
finitely many simple permutations.
Corollary
For all r and s, every subclass of Av(2413≤r , 3142≤s) containsonly finitely many simple permutations and thus has analgebraic generating function.
![Page 59: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/59.jpg)
A Decomposition Theorem with Enumerative Consequences
Pin Sequences
Proper Pin Sequences
Start with any two points.
Extend up, down, left, or right – this is a right pin.
A proper pin must be maximal and cut the previous pin, butnot the rectangle.
A right-reaching pin sequence.
![Page 60: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/60.jpg)
A Decomposition Theorem with Enumerative Consequences
Pin Sequences
Proper Pin Sequences
Start with any two points.
Extend up, down, left, or right – this is a right pin.
A proper pin must be maximal and cut the previous pin, butnot the rectangle.
A right-reaching pin sequence.
![Page 61: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/61.jpg)
A Decomposition Theorem with Enumerative Consequences
Pin Sequences
Proper Pin Sequences
Start with any two points.
Extend up, down, left, or right – this is a right pin.
A proper pin must be maximal and cut the previous pin, butnot the rectangle.
A right-reaching pin sequence.
![Page 62: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/62.jpg)
A Decomposition Theorem with Enumerative Consequences
Pin Sequences
Proper Pin Sequences
Start with any two points.
Extend up, down, left, or right – this is a right pin.
A proper pin must be maximal and cut the previous pin, butnot the rectangle.
A right-reaching pin sequence.
![Page 63: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/63.jpg)
A Decomposition Theorem with Enumerative Consequences
Pin Sequences
Proper Pin Sequences
Start with any two points.
Extend up, down, left, or right – this is a right pin.
A proper pin must be maximal and cut the previous pin, butnot the rectangle.
A right-reaching pin sequence.
![Page 64: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/64.jpg)
A Decomposition Theorem with Enumerative Consequences
Pin Sequences
Proper Pin Sequences
Start with any two points.
Extend up, down, left, or right – this is a right pin.
A proper pin must be maximal and cut the previous pin, butnot the rectangle.
A right-reaching pin sequence.
![Page 65: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/65.jpg)
A Decomposition Theorem with Enumerative Consequences
Pin Sequences
Proper Pin Sequences
Start with any two points.
Extend up, down, left, or right – this is a right pin.
A proper pin must be maximal and cut the previous pin, butnot the rectangle.
A right-reaching pin sequence.
![Page 66: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/66.jpg)
A Decomposition Theorem with Enumerative Consequences
Pin Sequences
Proper Pin Sequences
Start with any two points.
Extend up, down, left, or right – this is a right pin.
A proper pin must be maximal and cut the previous pin, butnot the rectangle.
A right-reaching pin sequence.
![Page 67: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/67.jpg)
A Decomposition Theorem with Enumerative Consequences
Pin Sequences
Simples and Pin Sequences
The points of the proper pin sequence form a simplepermutation.
![Page 68: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/68.jpg)
A Decomposition Theorem with Enumerative Consequences
Decomposing Simple Permutations
A Technical Theorem
Theorem
Every simple permutation of length at least 2(2048k8)(2048k8)(2k)
contains either a proper pin sequence of length at least 2k or aparallel alternation or a wedge simple permutation of length atleast 2k.
Proper pin sequence ⇒ two almost disjoint simples.
Parallel alternation ⇒ two almost disjoint simples.
Wedge simple permutation ⇒ two almost disjoint simples.
![Page 69: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/69.jpg)
A Decomposition Theorem with Enumerative Consequences
Decomposing Simple Permutations
A Technical Theorem
Theorem
Every simple permutation of length at least 2(2048k8)(2048k8)(2k)
contains either a proper pin sequence of length at least 2k or aparallel alternation or a wedge simple permutation of length atleast 2k.
Proper pin sequence ⇒ two almost disjoint simples.
Parallel alternation ⇒ two almost disjoint simples.
Wedge simple permutation ⇒ two almost disjoint simples.
![Page 70: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/70.jpg)
A Decomposition Theorem with Enumerative Consequences
Decomposing Simple Permutations
A Technical Theorem
Theorem
Every simple permutation of length at least 2(2048k8)(2048k8)(2k)
contains either a proper pin sequence of length at least 2k or aparallel alternation or a wedge simple permutation of length atleast 2k.
Proper pin sequence ⇒ two almost disjoint simples.
Parallel alternation ⇒ two almost disjoint simples.
Wedge simple permutation ⇒ two almost disjoint simples.
![Page 71: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/71.jpg)
A Decomposition Theorem with Enumerative Consequences
Decomposing Simple Permutations
A Technical Theorem
Theorem
Every simple permutation of length at least 2(2048k8)(2048k8)(2k)
contains either a proper pin sequence of length at least 2k or aparallel alternation or a wedge simple permutation of length atleast 2k.
Proper pin sequence ⇒ two almost disjoint simples.
Parallel alternation ⇒ two almost disjoint simples.
Wedge simple permutation ⇒ two almost disjoint simples.
![Page 72: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/72.jpg)
A Decomposition Theorem with Enumerative Consequences
Decomposing Simple Permutations
The Decomposition Theorem
Theorem (RB, SH, VV)
There is a function f (k) such that every simple permutation oflength at least f (k) contains two simple subsequences, each oflength at least k, which share at most two entries in common.
![Page 73: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/73.jpg)
Decidability and Unavoidable Structures
Outline
1 Basic ConceptsPermutation ClassesIntervals and Simple Permutations
2 Algebraic Generating Functions for Sets of PermutationsFinitely Many SimplesSets of Permutations
3 A Decomposition Theorem with Enumerative ConsequencesAimPin SequencesDecomposing Simple Permutations
4 Decidability and Unavoidable StructuresMore on PinsDecidability
![Page 74: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/74.jpg)
Decidability and Unavoidable Structures
More on Pins
The Language of Pins
Encode as: 1
Pattern involvement ↔ partial order on pin words.
Avoiding a pattern ↔ avoiding every pin word generatingthat pattern.
![Page 75: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/75.jpg)
Decidability and Unavoidable Structures
More on Pins
The Language of Pins
Encode as: 11
Pattern involvement ↔ partial order on pin words.
Avoiding a pattern ↔ avoiding every pin word generatingthat pattern.
![Page 76: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/76.jpg)
Decidability and Unavoidable Structures
More on Pins
The Language of Pins
Encode as: 11R
Pattern involvement ↔ partial order on pin words.
Avoiding a pattern ↔ avoiding every pin word generatingthat pattern.
![Page 77: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/77.jpg)
Decidability and Unavoidable Structures
More on Pins
The Language of Pins
Encode as: 11RU
Pattern involvement ↔ partial order on pin words.
Avoiding a pattern ↔ avoiding every pin word generatingthat pattern.
![Page 78: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/78.jpg)
Decidability and Unavoidable Structures
More on Pins
The Language of Pins
Encode as: 11RUL
Pattern involvement ↔ partial order on pin words.
Avoiding a pattern ↔ avoiding every pin word generatingthat pattern.
![Page 79: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/79.jpg)
Decidability and Unavoidable Structures
More on Pins
The Language of Pins
Encode as: 11RULD
Pattern involvement ↔ partial order on pin words.
Avoiding a pattern ↔ avoiding every pin word generatingthat pattern.
![Page 80: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/80.jpg)
Decidability and Unavoidable Structures
More on Pins
The Language of Pins
Encode as: 11RULDR
Pattern involvement ↔ partial order on pin words.
Avoiding a pattern ↔ avoiding every pin word generatingthat pattern.
![Page 81: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/81.jpg)
Decidability and Unavoidable Structures
More on Pins
The Language of Pins
Encode as: 11RULDRU
Pattern involvement ↔ partial order on pin words.
Avoiding a pattern ↔ avoiding every pin word generatingthat pattern.
![Page 82: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/82.jpg)
Decidability and Unavoidable Structures
More on Pins
The Language of Pins
Encode as: 11RULDRU
Pattern involvement ↔ partial order on pin words.
Avoiding a pattern ↔ avoiding every pin word generatingthat pattern.
![Page 83: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/83.jpg)
Decidability and Unavoidable Structures
More on Pins
The Language of Pins
Encode as: 11RULDRU
Pattern involvement ↔ partial order on pin words.
Avoiding a pattern ↔ avoiding every pin word generatingthat pattern.
![Page 84: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/84.jpg)
Decidability and Unavoidable Structures
Decidability
Decidability
Theorem (RB, NR, VV)
It is decidable whether a finitely based permutation classcontains only finitely many simple permutations.
Proof.
Technical theorem ⇒ only look for arbitrary parallel orwedge simple permutations, or proper pin sequences.
Parallel and wedge simple permutations easily verified.
Proper pin sequences ↔ the language of pins.
Language of pins avoiding a given pattern is regular.
Decidable if a regular language is infinite.
![Page 85: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/85.jpg)
Decidability and Unavoidable Structures
Decidability
Decidability
Theorem (RB, NR, VV)
It is decidable whether a finitely based permutation classcontains only finitely many simple permutations.
Proof.
Technical theorem ⇒ only look for arbitrary parallel orwedge simple permutations, or proper pin sequences.
Parallel and wedge simple permutations easily verified.
Proper pin sequences ↔ the language of pins.
Language of pins avoiding a given pattern is regular.
Decidable if a regular language is infinite.
![Page 86: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/86.jpg)
Decidability and Unavoidable Structures
Decidability
Decidability
Theorem (RB, NR, VV)
It is decidable whether a finitely based permutation classcontains only finitely many simple permutations.
Proof.
Technical theorem ⇒ only look for arbitrary parallel orwedge simple permutations, or proper pin sequences.
Parallel and wedge simple permutations easily verified.
Proper pin sequences ↔ the language of pins.
Language of pins avoiding a given pattern is regular.
Decidable if a regular language is infinite.
![Page 87: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/87.jpg)
Decidability and Unavoidable Structures
Decidability
Decidability
Theorem (RB, NR, VV)
It is decidable whether a finitely based permutation classcontains only finitely many simple permutations.
Proof.
Technical theorem ⇒ only look for arbitrary parallel orwedge simple permutations, or proper pin sequences.
Parallel and wedge simple permutations easily verified.
Proper pin sequences ↔ the language of pins.
Language of pins avoiding a given pattern is regular.
Decidable if a regular language is infinite.
![Page 88: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/88.jpg)
Decidability and Unavoidable Structures
Decidability
Decidability
Theorem (RB, NR, VV)
It is decidable whether a finitely based permutation classcontains only finitely many simple permutations.
Proof.
Technical theorem ⇒ only look for arbitrary parallel orwedge simple permutations, or proper pin sequences.
Parallel and wedge simple permutations easily verified.
Proper pin sequences ↔ the language of pins.
Language of pins avoiding a given pattern is regular.
Decidable if a regular language is infinite.
![Page 89: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/89.jpg)
Decidability and Unavoidable Structures
Decidability
Decidability
Theorem (RB, NR, VV)
It is decidable whether a finitely based permutation classcontains only finitely many simple permutations.
Proof.
Technical theorem ⇒ only look for arbitrary parallel orwedge simple permutations, or proper pin sequences.
Parallel and wedge simple permutations easily verified.
Proper pin sequences ↔ the language of pins.
Language of pins avoiding a given pattern is regular.
Decidable if a regular language is infinite.
![Page 90: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/90.jpg)
Decidability and Unavoidable Structures
Decidability
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![Page 91: Simple Permutations](https://reader031.vdocuments.site/reader031/viewer/2022013012/61cebe3ed61f3417bf20571d/html5/thumbnails/91.jpg)
Decidability and Unavoidable Structures
Decidability
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