Beautiful Bijections for Permutation Lara Pudwell Patterns - - PowerPoint PPT Presentation

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Beautiful Bijections for Permutation Lara Pudwell Patterns - - PowerPoint PPT Presentation

Feb 14, 2023 121 likes •573 views

Beautiful Bijections for Permutation Patterns Beautiful Bijections for Permutation Lara Pudwell Patterns Pattern- Avoiding Permutations Strategy Lara Pudwell Beautiful Valparaiso University Bijections Compositions Dyck Paths Others?

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Beautiful Bijections for Permutation Patterns

Lara Pudwell Valparaiso University Joint Mathematics Meetings MAA Invited Paper Session on Clever Counting or Beautiful Bijection January 5, 2012

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Outline

1

Pattern-Avoiding Permutations

2

Strategy

3

Beautiful Bijections Compositions Dyck Paths Others? ...When All Else Fails

4

Summary

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations A permutation of length n is an ordered list of the numbers {1, 2, . . . , n}. There are n! permutations of length n. Example: the 6 permutations of {1, 2, 3} are 123, 132, 213, 231, 312, 321.

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Graphs of Permutations Consider the permutation π = π1π2 · · · πn as a function from {1, 2, . . . , n} to {1, 2, . . . , n}. Example, π = 51342 π1 = 5 π2 = 1 π3 = 3 π4 = 4 π5 = 2

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21, 123,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21, 123, 132,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21, 123, 132, 231,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21, 123, 132, 231, 312,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21, 123, 132, 231, 312, 321,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21, 123, 132, 231, 312, 321, 1342,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21, 123, 132, 231, 312, 321, 1342, 4123,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21, 123, 132, 231, 312, 321, 1342, 4123, 4132,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21, 123, 132, 231, 312, 321, 1342, 4123, 4132, 4231,

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21, 123, 132, 231, 312, 321, 1342, 4123, 4132, 4231, 51342.

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations Inside Permutations Containment/Avoidance π = π1 · · · πn contains ρ = ρ1 · · · ρk as a pattern if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that πia < πib if and only if ρa < ρb. Otherwise π avoids the pattern ρ. Picture definition: π = 51342 contains 1, 12, 21, 123, 132, 231, 312, 321, 1342, 4123, 4132, 4231, 51342. π = 51342 avoids all other permutations.

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

A Family of Counting Problems Sn(Q) Sn(Q) is the set of permutations of length n that avoid all permutations in Q. sn(Q) sn(Q) = |Sn(Q)| Problem Given a list of permutations Q, describe the structure of π ∈ Sn(Q) and/or find an expression for sn(Q). (Generally Hard)

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Strategy

1

Count (Find sn(Q) for as many Q as possible.)

2

Compare (Where have I seen this sequence before?)

3

Connect (Why did I get the same sequence twice?)

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Strategy

1

Count (Find sn(Q) for as many Q as possible.)

2

Compare (Where have I seen this sequence before?)

3

Connect (Why did I get the same sequence twice?) Argument 1 For a family of counting problems... clever counting says “these things are the same”, but beautiful bijections explain “why things are the same”. Argument 2 Beautiful bijections allow us to translate what we know about

  • ne object to better understand another object.
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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations and Compositions Theorem sn(123, 132) is equal to the number of compositions of n. Recall: a composition of n is an ordered list of positive integers whose sum is n. Example: Members of S4(123, 132) are: 3214, 3241, 3412, 3421, 4213, 4231, 4312, 4321 Compositions of 4 are: 4, 1 + 3, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 2 + 1, 1 + 1 + 2, 1 + 1 + 1 + 1

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Structure of a permutation in Sn(123, 132)

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Structure of a permutation in Sn(123, 132)

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Structure of a permutation in Sn(123, 132)

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Structure of a permutation in Sn(123, 132)

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Structure of a permutation in Sn(123, 132)

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Structure of a permutation in Sn(123, 132)

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Structure of a permutation in Sn(123, 132) → 5 + 2 + 4 + 1

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations and Dyck Paths Theorem sn(321) is equal to the number of Dyck paths of length 2n. Recall: a Dyck path of length 2n is a path in the xy-plane from (0, 0) to (n, n) that (a) only uses the steps 1, 0 and 0, 1 and (b) never goes above the line y = x.

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Permutations and Dyck Paths Theorem sn(321) is equal to the number of Dyck paths of length 2n. Recall: a (shifted) Dyck path of length 2n is a path in the xy-plane from (1, 0) to (n + 1, n) that (a) only uses the steps 1, 0 and 0, 1 and (b) never goes above the line y = x − 1.

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

From π ∈ Sn(321) to a (shifted) Dyck path

1

Delete all left-to-right maxima

2

Add the point (n + 1, n).

3

Start at (1, 0). Move right until under a point, then up to the point, and repeat.

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

From π ∈ Sn(321) to a (shifted) Dyck path

1

Delete all left-to-right maxima

2

Add the point (n + 1, n).

3

Start at (1, 0). Move right until under a point, then up to the point, and repeat.

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

From π ∈ Sn(321) to a (shifted) Dyck path

1

Delete all left-to-right maxima

2

Add the point (n + 1, n).

3

Start at (1, 0). Move right until under a point, then up to the point, and repeat.

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

From π ∈ Sn(321) to a (shifted) Dyck path

1

Delete all left-to-right maxima

2

Add the point (n + 1, n).

3

Start at (1, 0). Move right until under a point, then up to the point, and repeat.

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

From π ∈ Sn(321) to a (shifted) Dyck path

1

Delete all left-to-right maxima

2

Add the point (n + 1, n).

3

Start at (1, 0). Move right until under a point, then up to the point, and repeat. Notice:

1

All non-left-to-right-maxima are in increasing order.

2

For any point non-left-to-right-maxima (i, πi), there are at least 1 + (πi − 1) = πi points to the left of (i, πi), so i ≥ πi + 1, or πi ≤ i − 1, as desired.

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Other Bijections There exist bijections between various sets of pattern-avoiding permutations and... set partitions, trees, faces in certain geometric solids, ... and more!

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Symmetry Bijections Argument 3 There are times when clever counting fails, but beautiful bijections succeed!

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Symmetry Bijections Argument 3 There are times when clever counting fails, but beautiful bijections succeed!

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Symmetry Bijections Argument 3 There are times when clever counting fails, but beautiful bijections succeed! sn(51342) = sn(24315) = sn(15324) = sn(25341)

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

The Recap While counting can be classy, bijections have some advantages. In particular... bijections explain “why”. bijections allow knowledge about one object to help discover new properties of another object. bijections may succeed even when counting fails!

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Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding Permutations Strategy Beautiful Bijections

Compositions Dyck Paths Others? ...When All Else Fails

Summary

Thank you for listening!