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Counting Pattern-Avoiding Permutations Lara Pudwell Valparaiso - - PowerPoint PPT Presentation

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Counting Pattern-Avoiding Permutations Lara Pudwell Valparaiso University Trinity University Mathematics Majors

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems

Counting Pattern-Avoiding Permutations

Lara Pudwell

Valparaiso University

Trinity University Mathematics Majors’ Seminar November 19, 2009

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems

Outline

1

What is a pattern-avoiding permutation?

2

Enumeration

3

Motivational Interlude

4

Variations of Pattern Avoidance

5

Open Problems

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Permutations

A permutation of length n is a string of numbers using each of 1, . . . , n exactly once.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 4

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Permutations

A permutation of length n is a string of numbers using each of 1, . . . , n exactly once. Write Sn for the set of permutations of length n. For example: S1 = {1} S2 = {12, 21} S3 = {123, 132, 213, 231, 312, 321}

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Reduction

Given a string of numbers q = q1 · · · qm, the reduction of q is the string obtained by replacing the ith smallest number of q with i. For example, the reduction of 26745 is 14523.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance/Containment

Given permutations p = p1 · · · pn and q = q1 · · · qm, p contains q as a pattern if there is 1 ≤ i1 < · · · < im ≤ n so that pi1 · · · pim reduces to q;

  • therwise p avoids q.

For example, 4576213 contains 312 (4576213). 4576213 avoids 1234.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Permutations as Functions

We can also think of a permutation as a function from {1, . . . , n} to {1, . . . , n}. p = 4576213

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Permutations as Functions

We can also think of a permutation as a function from {1, . . . , n} to {1, . . . , n}. p = 4576213 Then, permutation p contains permutation q if the graph of p contains the graph of q. 4576213 contains 312.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 9

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 10

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the pattern 1.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 11

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the pattern 1.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 12

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the pattern 1.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 13

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the pattern 1.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 14

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the patterns 1, 12.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 15

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the patterns 1, 12.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 16

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the patterns 1, 12.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 17

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the patterns 1, 12, 21.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 18

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the patterns 1, 12, 21.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 19

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the patterns 1, 12, 21.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 20

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the patterns 1, 12, 21, 132.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 21

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the patterns 1, 12, 21, 132.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 22

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the patterns 1, 12, 21, 132, 123.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 23

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the patterns 1, 12, 21, 132, 123, 312.

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 24

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Easier Question: Fix p. What patterns are contained in p? For example, p = 1423 1423 contains the patterns 1, 12, 21, 132, 123, 312, and 1423, and avoids all other patterns.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Pattern Avoidance Definitions Main Questions

Pattern Avoidance in Permutations

Harder Question: Fix a pattern q. Enumerate Sn(q) := {p ∈ Sn | p avoids q}.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 1

There is one pattern of length 1: 1. What is |Sn(1)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 1

There is one pattern of length 1: 1. What is |Sn(1)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 28

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 1

There is one pattern of length 1: 1. What is |Sn(1)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 1

There is one pattern of length 1: 1. What is |Sn(1)|? |Sn(1)| =

  • 1

n = 0 n ≥ 1.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 2

There are two patterns of length 2: 12, 21. What is |Sn(12)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 31

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 2

There are two patterns of length 2: 12, 21. What is |Sn(12)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 32

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 2

There are two patterns of length 2: 12, 21. What is |Sn(12)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 33

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 2

There are two patterns of length 2: 12, 21. What is |Sn(12)|? |Sn(12)| = 1 (for n ≥ 0).

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 2

There are two patterns of length 2: 12, 21. What is |Sn(12)|? What is |Sn(21)|? |Sn(12)| = 1 (for n ≥ 0).

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 2

There are two patterns of length 2: 12, 21. What is |Sn(12)|? What is |Sn(21)|? |Sn(12)| = 1 (for n ≥ 0).

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 2

There are two patterns of length 2: 12, 21. What is |Sn(12)|? What is |Sn(21)|? |Sn(12)| = 1 (for n ≥ 0).

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 2

There are two patterns of length 2: 12, 21. What is |Sn(12)|? What is |Sn(21)|? |Sn(12)| = |Sn(21)| = 1 (for n ≥ 0).

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Useful Observation (Wilf Equivalence)

q (any pattern) qr (reverse) qc (complement) q−1 (inverse) For any pattern q, we have: |Sn(q)| = |Sn(qr)| = |Sn(qc)| =

  • Sn(q−1)
  • Lara Pudwell

Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 3

There are six patterns of length 3: 123, 132, 213, 231, 312, 321. Using the useful observation, we have |Sn(123)| = |Sn(321)| and |Sn(132)| = |Sn(231)| = |Sn(213)| = |Sn(312)|.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 3

There are six patterns of length 3: 123, 132, 213, 231, 312, 321. Using the useful observation, we have |Sn(123)| = |Sn(321)| and |Sn(132)| = |Sn(231)| = |Sn(213)| = |Sn(312)|. |Sn(123)| = |Sn(132)| (Simion and Schmidt, 1985).

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding the pattern 132

What is |Sn(132)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 42

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding the pattern 132

What is |Sn(132)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 43

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding the pattern 132

What is |Sn(132)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 44

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding the pattern 132

What is |Sn(132)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 45

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding the pattern 132

What is |Sn(132)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 46

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding the pattern 132

What is |Sn(132)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 47

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding the pattern 132

What is |Sn(132)|?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 48

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding the pattern 132

What is |Sn(132)|? |Sn(132)| =

n

  • i=1

|Si−1(132)| · |Sn−i(132)| (for n > 0)

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 49

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding the pattern 132

What is |Sn(132)|? |Sn(132)| =

n

  • i=1

|Si−1(132)| · |Sn−i(132)| (for n > 0) |Sn(132)| = 2n

n

  • n + 1 = nth Catalan number

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 4

There are 24 patterns of length 4. Using the useful observation and similar bijections, we can narrow our work to 3 cases: Sn(1342), Sn(1234), and Sn(1324).

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Length 1 Length 2 Length 3 Length 4

Avoiding a Pattern of Length 4

There are 24 patterns of length 4. Using the useful observation and similar bijections, we can narrow our work to 3 cases: Sn(1342), Sn(1234), and Sn(1324).

1 2 3 4 5 6 7 8 |Sn(1342)| 1 2 6 23 103 512 2740 15485 ∼ 8n |Sn(1234)| 1 2 6 23 103 513 2761 15767 ∼ 9n |Sn(1324)| 1 2 6 23 103 513 2762 15793 ∼ 9.3n

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Stack Sorting Schubert Varieties Experimental Mathematics

Stack Sorting

A stack is a last in, first out data structure.

132 32 32 2 1 → 1 → 1 → 3 → 2 3 → 1 1 1 1 123

Question: What permutations can be sorted by passing them through the stack exactly one time?

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 53

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Stack Sorting Schubert Varieties Experimental Mathematics

Stack Sorting

A stack is a last in, first out data structure.

132 32 32 2 1 → 1 → 1 → 3 → 2 3 → 1 1 1 1 123

Question: What permutations can be sorted by passing them through the stack exactly one time? Answer: (Knuth, 1973) Exactly the permutations that avoid 231.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Stack Sorting Schubert Varieties Experimental Mathematics

Schubert Varieties

1900: Hilbert’s 15th problem is to find a "rigorous foundation of Schubert’s enumerative calculus". 1900s: In algebraic geometry, Schubert varieties are the most commonly studied type of singular variety. Question: Which Schubert varieties are smooth?

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Stack Sorting Schubert Varieties Experimental Mathematics

Schubert Varieties

1900: Hilbert’s 15th problem is to find a "rigorous foundation of Schubert’s enumerative calculus". 1900s: In algebraic geometry, Schubert varieties are the most commonly studied type of singular variety. Question: Which Schubert varieties are smooth? Answer: (Lakshmibai and Sandhya, 1990): Exactly the varieties whose indexing permutation avoids 4231 and 3412.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Stack Sorting Schubert Varieties Experimental Mathematics

Experimental Mathematics

The techniques to count |Sn(q)| usually depend on q. Goal: Find an algorithm to count |Sn(q)| that works well regardless of what q is.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Avoiding Sets of Patterns Pattern-Avoiding Words Generalized and Distanced Patterns Barred Pattern Avoidance

Avoiding Sets of Patterns

We may consider permutations which simultaneously avoid more than one pattern: Sn(Q) := {p ∈ Sn | p avoids q for every q ∈ Q}. Some particularly nice results include: Sn({123, 132}) = 2n−1 Sn({132, 213, 321}) = n Sn({123, 132, 213}) = Fn (Fibonacci numbers)

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Avoiding Sets of Patterns Pattern-Avoiding Words Generalized and Distanced Patterns Barred Pattern Avoidance

Pattern-Avoiding Words

Instead of pattern-avoiding permutations, we may consider strings with repeated letters: A[a1,...,ak](Q) :=

  • w ∈ [k]

ai

  • w has ai i’s,

w avoids q for every q ∈ Q

  • Notice A[1,...,1](Q) = Sn(Q).

A[2,...,2]({132, 231, 2134}) = 2n2 + 6n − 2

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Avoiding Sets of Patterns Pattern-Avoiding Words Generalized and Distanced Patterns Barred Pattern Avoidance

Generalized and Distanced Patterns

Consider patterns where there may or may not be a dash between each pair of numbers. E.g. 3 − 12 A dash indicates those two numbers can be arbitrarily far apart, no dash indicates they must be adjacent. E.g. 241653 contains 12 − 3 (241653), but not 1 − 23 Further, we can specify exact distances between numbers of a pattern.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Avoiding Sets of Patterns Pattern-Avoiding Words Generalized and Distanced Patterns Barred Pattern Avoidance

Bar Notation

A barred permutation pattern is a permutation where each number may or may not have a bar over it. E.g. q = 31542 is a barred pattern.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Avoiding Sets of Patterns Pattern-Avoiding Words Generalized and Distanced Patterns Barred Pattern Avoidance

Bar Notation

A barred permutation pattern is a permutation where each number may or may not have a bar over it. E.g. q = 31542 is a barred pattern. A barred pattern q encodes two permutation patterns,

1

The smaller pattern qs formed by the unbarred numbers of q. (in this case, 542 forms a 321 pattern.)

2

The larger pattern qℓ formed by all numbers of q. (in this case, 31542.)

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Avoiding Sets of Patterns Pattern-Avoiding Words Generalized and Distanced Patterns Barred Pattern Avoidance

Bar Notation

We say that permutation p avoids the barred pattern q iff every copy of qs in p is part of a copy of qℓ in p. Example: q = 31542

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 63

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Avoiding Sets of Patterns Pattern-Avoiding Words Generalized and Distanced Patterns Barred Pattern Avoidance

Bar Notation

We say that permutation p avoids the barred pattern q iff every copy of qs in p is part of a copy of qℓ in p. Example: q = 31542

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 64

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Avoiding Sets of Patterns Pattern-Avoiding Words Generalized and Distanced Patterns Barred Pattern Avoidance

Bar Notation

We say that permutation p avoids the barred pattern q iff every copy of qs in p is part of a copy of qℓ in p. Example: q = 31542

Lara Pudwell Counting Pattern-Avoiding Permutations

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SLIDE 65

What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Avoiding Sets of Patterns Pattern-Avoiding Words Generalized and Distanced Patterns Barred Pattern Avoidance

Barred Pattern Avoidance

Two friendly examples: Sn(132) = (n − 1)! Sn(1423) = Bn (Bell numbers)

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems Avoiding Sets of Patterns Pattern-Avoiding Words Generalized and Distanced Patterns Barred Pattern Avoidance

Barred Pattern Avoidance

Two friendly examples: Sn(132) = (n − 1)! Sn(1423) = Bn (Bell numbers) Two Nice Theorems: West, 1990: A permutation is 2-stack sortable if and only if it avoids 2341 and 35241. Woo and Yong, 2006: A Schubert variety Xw is locally factorial if and only if w avoids the patterns 1324 and 21354.

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems

Open Problems

Enumeration

What is Sn(1324)?, Sn(q), where |q| ≥ 5? What can you say about permutations avoiding generalized

  • r barred patterns of length ≥ 5?

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems

Open Problems 2

Algebra

There are Wilf equivalences other than the symmetries of the square. What other equivalences can you find? What Wilf equivalences carry over to subsets or subgroups

  • f Sn?

Asymptotics

Given any pattern q, there exists a constant cq such that lim

n→∞ |Sn(q)| → cn q.

What values of cq are possible?

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems

Open Problems 3

Applications

(Stack Sorting) What is a characterization for 3-stack-sortable permutations? (Schubert calculus, etc.) Sn is an example of a Coxeter group (a group generated by reflections). What can you say about pattern avoidance in other Coxeter groups? (Experimental Mathematics) Can you find a single method that efficiently counts |Sn(Q)| for many different examples

  • f Q?

Lara Pudwell Counting Pattern-Avoiding Permutations

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What is a pattern-avoiding permutation? Enumeration Motivational Interlude Variations of Pattern Avoidance Open Problems

Thank You!

Lara Pudwell Counting Pattern-Avoiding Permutations