pattern avoidance classes invariant under the modified...
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Pattern Avoidance Classes Invariant Under the ModifiedFoata-Strehl Action
Richard Zhou
Mentor: Yan Zhuang
May 20, 2017MIT PRIMES Conference
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 1
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Permutations
Definition
A permutation of a the set {1, 2, 3, . . . , n} is a length-n string that useseach number from 1 to n once.
Example
The string π = 67284135 is a permutation of the set {1, 2, 3, 4, 5, 6, 7, 8}.
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Permutations
Definition
A permutation of a the set {1, 2, 3, . . . , n} is a length-n string that useseach number from 1 to n once.
Example
The string π = 67284135 is a permutation of the set {1, 2, 3, 4, 5, 6, 7, 8}.
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Peaks and Valleys
Imagine a permutation as a ”mountain range”.Each letter is either a peak, valley, or free letter.
Example
The ”mountain range” representation of the permutation 67284135:
.
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Peaks and Valleys
Imagine a permutation as a ”mountain range”.Each letter is either a peak, valley, or free letter.
Example
The ”mountain range” representation of the permutation 67284135:
.
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Pattern Avoidance
Consider a ”long” permutation, such as π = 23514 and a shorter one,say σ = 132.
The 3-tuple of entries (2,5,4) in π forms a pattern of type 132.
We say the permutation π contains σ.
The permutation 15234 avoids the pattern 321.
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Pattern Avoidance
Consider a ”long” permutation, such as π = 23514 and a shorter one,say σ = 132.
The 3-tuple of entries (2,5,4) in π forms a pattern of type 132.
We say the permutation π contains σ.
The permutation 15234 avoids the pattern 321.
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 4
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Pattern Avoidance
Consider a ”long” permutation, such as π = 23514 and a shorter one,say σ = 132.
The 3-tuple of entries (2,5,4) in π forms a pattern of type 132.
We say the permutation π contains σ.
The permutation 15234 avoids the pattern 321.
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 4
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Pattern Avoidance Classes
Let π be a permutation of [n] and let Σ = {σ1, σ2, . . . , σm} be a setof patterns each of length at most n. We say that π avoids Σ if πavoids every pattern in Σ.
Avn(Σ) denotes the set of all length-n permutations p such that pavoids Σ.
Av(Σ) denotes the set of all permutations that avoid Σ.
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Pattern Avoidance Classes
Let π be a permutation of [n] and let Σ = {σ1, σ2, . . . , σm} be a setof patterns each of length at most n. We say that π avoids Σ if πavoids every pattern in Σ.
Avn(Σ) denotes the set of all length-n permutations p such that pavoids Σ.
Av(Σ) denotes the set of all permutations that avoid Σ.
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Valley Hopping
Definition
A valley-hop (formally known as the modified Foata-Strehl Action) Hj(σ)is the permutation obtained by moving the free letter j in σ across theadjacent valleys to the nearest slope of the same height.
Example
Hj(σ) for j = 5 and σ = 67284135
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Hop Equivalence Classes
Valley-hopping naturally partitions the set of length-n permutationsinto equivalence classes:
Definition
Two permutations σ1 and σ2 of [n] are in the same hop equivalence classif there exists some sequence of valley-hops Hi1 ,Hi2 , . . . ,Hik such thatHi1(Hi2(...(Hik (σ1))...)) = σ2. We let Hop(σ) denote the hop equivalenceclass of σ.
Example
Hop(13542) = {13542, 13524, 31542, 31524}
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Hop Equivalence Classes
Valley-hopping naturally partitions the set of length-n permutationsinto equivalence classes:
Definition
Two permutations σ1 and σ2 of [n] are in the same hop equivalence classif there exists some sequence of valley-hops Hi1 ,Hi2 , . . . ,Hik such thatHi1(Hi2(...(Hik (σ1))...)) = σ2. We let Hop(σ) denote the hop equivalenceclass of σ.
Example
Hop(13542) = {13542, 13524, 31542, 31524}
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Hop Equivalence Classes
Valley-hopping naturally partitions the set of length-n permutationsinto equivalence classes:
Definition
Two permutations σ1 and σ2 of [n] are in the same hop equivalence classif there exists some sequence of valley-hops Hi1 ,Hi2 , . . . ,Hik such thatHi1(Hi2(...(Hik (σ1))...)) = σ2. We let Hop(σ) denote the hop equivalenceclass of σ.
Example
Hop(13542) = {13542, 13524, 31542, 31524}
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Valley Hopping and Pattern Avoidance Classes
Definition
Let Σ be a set of patterns. We say that Avn(Σ) is invariant undervalley-hopping if for any permutation π ∈ Avn(Σ), any valley-hop π′ of πis also in Avn(Σ).
If Avn(Σ) is invariant under valley-hopping for all n, the distributionof peaks and valleys for permutations in Av(Σ) is well understood.
Our problem: Classify all pattern sets Σ such that Av(Σ) is invariantunder valley-hopping.
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 8
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Valley Hopping and Pattern Avoidance Classes
Definition
Let Σ be a set of patterns. We say that Avn(Σ) is invariant undervalley-hopping if for any permutation π ∈ Avn(Σ), any valley-hop π′ of πis also in Avn(Σ).
If Avn(Σ) is invariant under valley-hopping for all n, the distributionof peaks and valleys for permutations in Av(Σ) is well understood.
Our problem: Classify all pattern sets Σ such that Av(Σ) is invariantunder valley-hopping.
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 8
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Valley Hopping and Pattern Avoidance Classes
Definition
Let Σ be a set of patterns. We say that Avn(Σ) is invariant undervalley-hopping if for any permutation π ∈ Avn(Σ), any valley-hop π′ of πis also in Avn(Σ).
If Avn(Σ) is invariant under valley-hopping for all n, the distributionof peaks and valleys for permutations in Av(Σ) is well understood.
Our problem: Classify all pattern sets Σ such that Av(Σ) is invariantunder valley-hopping.
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 8
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Singleton Patterns
A set of patterns Σ is called singleton if |Σ| = 1.
Known that if Σ = {132} or Σ = {231}, then Av(Σ) is invariantunder valley-hopping.
We show that these are the only possible values for Σ:
Proposition
Let Σ be a nontrivial singleton pattern set with Av(Σ) invariant undervalley-hopping. Then Σ = {132} or Σ = {231}.
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Singleton Patterns
A set of patterns Σ is called singleton if |Σ| = 1.
Known that if Σ = {132} or Σ = {231}, then Av(Σ) is invariantunder valley-hopping.
We show that these are the only possible values for Σ:
Proposition
Let Σ be a nontrivial singleton pattern set with Av(Σ) invariant undervalley-hopping. Then Σ = {132} or Σ = {231}.
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 9
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Singleton Patterns
A set of patterns Σ is called singleton if |Σ| = 1.
Known that if Σ = {132} or Σ = {231}, then Av(Σ) is invariantunder valley-hopping.
We show that these are the only possible values for Σ:
Proposition
Let Σ be a nontrivial singleton pattern set with Av(Σ) invariant undervalley-hopping. Then Σ = {132} or Σ = {231}.
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 9
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Singleton Patterns
A set of patterns Σ is called singleton if |Σ| = 1.
Known that if Σ = {132} or Σ = {231}, then Av(Σ) is invariantunder valley-hopping.
We show that these are the only possible values for Σ:
Proposition
Let Σ be a nontrivial singleton pattern set with Av(Σ) invariant undervalley-hopping. Then Σ = {132} or Σ = {231}.
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Single Hop Equivalence Classes
If Σ is a pattern set such that Av(Σ) is invariant under valley-hoppingand σ ∈ Σ, then Hop(σ) ⊆ Σ.
Only nontrivial Σ with more than one element for which Av(Σ) wasknown to be invariant under valley-hopping was Σ = {1423, 1432}.Can we classify all σ for which Av(Hop(σ)) is invariant undervalley-hopping?
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Single Hop Equivalence Classes
If Σ is a pattern set such that Av(Σ) is invariant under valley-hoppingand σ ∈ Σ, then Hop(σ) ⊆ Σ.
Only nontrivial Σ with more than one element for which Av(Σ) wasknown to be invariant under valley-hopping was Σ = {1423, 1432}.
Can we classify all σ for which Av(Hop(σ)) is invariant undervalley-hopping?
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 10
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Single Hop Equivalence Classes
If Σ is a pattern set such that Av(Σ) is invariant under valley-hoppingand σ ∈ Σ, then Hop(σ) ⊆ Σ.
Only nontrivial Σ with more than one element for which Av(Σ) wasknown to be invariant under valley-hopping was Σ = {1423, 1432}.Can we classify all σ for which Av(Hop(σ)) is invariant undervalley-hopping?
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 10
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Classification for Single Hop Equivalence Classes
Single hop equivalence classes invariant under valley-hopping havespecific structure:
Proposition
Let σ be a permutation such that Av(Hop(σ)) is nontrivial and invariantunder valley-hopping. Then σ has exactly one peak, which is its largestletter.
Proposition
There does not exist a position i < |σ| for which i and i + 1 are both freeletters in σ.
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Classification for Single Hop Equivalence Classes
Single hop equivalence classes invariant under valley-hopping havespecific structure:
Proposition
Let σ be a permutation such that Av(Hop(σ)) is nontrivial and invariantunder valley-hopping. Then σ has exactly one peak, which is its largestletter.
Proposition
There does not exist a position i < |σ| for which i and i + 1 are both freeletters in σ.
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Classification for Single Hop Equivalence Classes
Single hop equivalence classes invariant under valley-hopping havespecific structure:
Proposition
Let σ be a permutation such that Av(Hop(σ)) is nontrivial and invariantunder valley-hopping. Then σ has exactly one peak, which is its largestletter.
Proposition
There does not exist a position i < |σ| for which i and i + 1 are both freeletters in σ.
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Classification for Single Hop Equivalence Classes (Cont.)
These properties place tight restrictions on σ:
Theorem
There are 14 nontrivial hop equivalence classes H for which Avn(H) isinvariant under valley hopping.
Av of the hop equivalence classes for following permutations are invariantunder valley-hopping:
132, 231
1423, 2413, 3412, 1243, 1342, 2341
12534, 13524, 14523, 23514, 24513, 34512
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Classification for Single Hop Equivalence Classes (Cont.)
These properties place tight restrictions on σ:
Theorem
There are 14 nontrivial hop equivalence classes H for which Avn(H) isinvariant under valley hopping.
Av of the hop equivalence classes for following permutations are invariantunder valley-hopping:
132, 231
1423, 2413, 3412, 1243, 1342, 2341
12534, 13524, 14523, 23514, 24513, 34512
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Classification for Single Hop Equivalence Classes (Cont.)
These properties place tight restrictions on σ:
Theorem
There are 14 nontrivial hop equivalence classes H for which Avn(H) isinvariant under valley hopping.
Av of the hop equivalence classes for following permutations are invariantunder valley-hopping:
132, 231
1423, 2413, 3412, 1243, 1342, 2341
12534, 13524, 14523, 23514, 24513, 34512
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Construction for General Pattern Sets
For any permutation pattern σ of length n, there is a trivial patternset Σ such that Avn(Σ) is invariant under valley-hopping. We wouldlike to improve upon this:
Theorem
Let σ be a permutation pattern of length n. Then there exists a family oflength-n permutation patterns Σ containing σ such that
|Σ| < n!
(n − pk(σ))!2n−2pk(σ)−1
where pk(σ) denotes the number of peaks in σ.
Improvement over trivial family size of n! by factor of (n−pk(σ))!2n−2pk(σ)−1 .
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 13
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Construction for General Pattern Sets
For any permutation pattern σ of length n, there is a trivial patternset Σ such that Avn(Σ) is invariant under valley-hopping. We wouldlike to improve upon this:
Theorem
Let σ be a permutation pattern of length n. Then there exists a family oflength-n permutation patterns Σ containing σ such that
|Σ| < n!
(n − pk(σ))!2n−2pk(σ)−1
where pk(σ) denotes the number of peaks in σ.
Improvement over trivial family size of n! by factor of (n−pk(σ))!2n−2pk(σ)−1 .
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 13
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![Page 33: Pattern Avoidance Classes Invariant Under the Modified ...math.mit.edu/research/highschool/primes/materials/2017/conf/5-1-Zhou.pdfRichard Zhou (Mentor: Yan Zhuang) Pattern Avoidance](https://reader035.vdocuments.us/reader035/viewer/2022071500/611fa5ff90ed5a28535382f4/html5/thumbnails/33.jpg)
Construction for General Pattern Sets
For any permutation pattern σ of length n, there is a trivial patternset Σ such that Avn(Σ) is invariant under valley-hopping. We wouldlike to improve upon this:
Theorem
Let σ be a permutation pattern of length n. Then there exists a family oflength-n permutation patterns Σ containing σ such that
|Σ| < n!
(n − pk(σ))!2n−2pk(σ)−1
where pk(σ) denotes the number of peaks in σ.
Improvement over trivial family size of n! by factor of (n−pk(σ))!2n−2pk(σ)−1 .
Richard Zhou (Mentor: Yan Zhuang) Pattern AvoidanceMay 20, 2017 MIT PRIMES Conference 13
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Alternating Permutations
Definition
A permutation is strictly alternating if it has no free letters.
Example
1745263 is a strictly alternating permutation.
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Families of Strictly Alternating Permutations
Definition
We call a permutation tall if every peak is greater than every valley.
Theorem
Let Σ be a family of strictly alternating permutations with k peaks suchthat Av(Σ) invariant under valley-hopping. Then there exists some subsetΠ of Σ of size (k − 1)! such that
Every permutation in Π is tall
Every permutation in Π has the same valleys
For any π ∈ Π, the letter in position 2 is the smallest peak.
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Families of Strictly Alternating Permutations
Definition
We call a permutation tall if every peak is greater than every valley.
Theorem
Let Σ be a family of strictly alternating permutations with k peaks suchthat Av(Σ) invariant under valley-hopping. Then there exists some subsetΠ of Σ of size (k − 1)! such that
Every permutation in Π is tall
Every permutation in Π has the same valleys
For any π ∈ Π, the letter in position 2 is the smallest peak.
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Future Work
Classify all sets of strictly alternating permutations with Av(Σ)invariant under valley-hopping.
Current strategy: start with singleton pattern set Σ and insert moreelements until Av(Σ) is invariant under valley-hopping. Is there amore general way of generating Σ invariant under valley-hopping?
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Acknowledgements
Many thanks to:
Yan Zhuang, my mentor
Dr. Tanya Khovanova
My parents
MIT PRIMES
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