Schemes for Pattern-Avoiding Words Lara Pudwell Rutgers University - - PowerPoint PPT Presentation

▶
schemes for pattern avoiding words
SMART_READER_LITE
LIVE PREVIEW

Schemes for Pattern-Avoiding Words Lara Pudwell Rutgers University - - PowerPoint PPT Presentation

Jun 21, 2023 184 likes •1.06k views

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Pattern-Avoiding Words Lara Pudwell Rutgers University Permutation Patterns 2007 Lara Pudwell Schemes for Pattern-Avoiding Words Introduction/History

slide-1
SLIDE 1

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary

Schemes for Pattern-Avoiding Words

Lara Pudwell

Rutgers University

Permutation Patterns 2007

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-2
SLIDE 2

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary

Outline

1

Introduction/History Pattern Avoidance in Words Previous Work

2

Prefix Schemes for Words Definitions Examples Success Rate

3

Other Schemes for Words Schemes for Monotone Patterns

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-3
SLIDE 3

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Outline

1

Introduction/History Pattern Avoidance in Words Previous Work

2

Prefix Schemes for Words Definitions Examples Success Rate

3

Other Schemes for Words Schemes for Monotone Patterns

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-4
SLIDE 4

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Reduction

Given a string of letters p = p1...pn, the reduction of p is the string obtained by replacing the ith smallest letter(s) of p with i.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-5
SLIDE 5

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Reduction

Given a string of letters p = p1...pn, the reduction of p is the string obtained by replacing the ith smallest letter(s) of p with i. For example, the reduction of 2674425 is

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-6
SLIDE 6

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Reduction

Given a string of letters p = p1...pn, the reduction of p is the string obtained by replacing the ith smallest letter(s) of p with i. For example, the reduction of 2674425 is 1• • ••1•.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-7
SLIDE 7

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Reduction

Given a string of letters p = p1...pn, the reduction of p is the string obtained by replacing the ith smallest letter(s) of p with i. For example, the reduction of 2674425 is 1••221•.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-8
SLIDE 8

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Reduction

Given a string of letters p = p1...pn, the reduction of p is the string obtained by replacing the ith smallest letter(s) of p with i. For example, the reduction of 2674425 is 1••2213.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-9
SLIDE 9

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Reduction

Given a string of letters p = p1...pn, the reduction of p is the string obtained by replacing the ith smallest letter(s) of p with i. For example, the reduction of 2674425 is 14•2213.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-10
SLIDE 10

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Reduction

Given a string of letters p = p1...pn, the reduction of p is the string obtained by replacing the ith smallest letter(s) of p with i. For example, the reduction of 2674425 is 1452213.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-11
SLIDE 11

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Pattern Avoidance in Words

Given w ∈ [k]n, and p = p1 . . . pm, w contains p if there is 1 ≤ i1 < · · · < im ≤ n so that wi1 . . . wim reduces to p. Otherwise w avoids p.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-12
SLIDE 12

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Pattern Avoidance in Words

Given w ∈ [k]n, and p = p1 . . . pm, w contains p if there is 1 ≤ i1 < · · · < im ≤ n so that wi1 . . . wim reduces to p. Otherwise w avoids p. e.g. 1452213 contains 312 (1452213) 1452213 avoids 212.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-13
SLIDE 13

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Pattern Avoidance in Words

Given w ∈ [k]n, and p = p1 . . . pm, w contains p if there is 1 ≤ i1 < · · · < im ≤ n so that wi1 . . . wim reduces to p. Otherwise w avoids p. e.g. 1452213 contains 312 (1452213) 1452213 avoids 212. Want to count A[a1,...,ak]({Q}) := {w ∈ [k]

ai | w has ai i’s, w avoids q for every q ∈ Q}

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-14
SLIDE 14

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Outline

1

Introduction/History Pattern Avoidance in Words Previous Work

2

Prefix Schemes for Words Definitions Examples Success Rate

3

Other Schemes for Words Schemes for Monotone Patterns

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-15
SLIDE 15

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Previous Work for Words

Results by...

Burstein: initial results, generating functions

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-16
SLIDE 16

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Previous Work for Words

Results by...

Burstein: initial results, generating functions Albert, Aldred, Atkinson, Handley, Holton: results for specific 3-letter patterns

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-17
SLIDE 17

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Previous Work for Words

Results by...

Burstein: initial results, generating functions Albert, Aldred, Atkinson, Handley, Holton: results for specific 3-letter patterns Brändén, Mansour: automata for enumeration, for specific k

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-18
SLIDE 18

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Previous Work for Words

Results by...

Burstein: initial results, generating functions Albert, Aldred, Atkinson, Handley, Holton: results for specific 3-letter patterns Brändén, Mansour: automata for enumeration, for specific k

Note: most work is for specific patterns, would like a universal technique that works well regardless of pattern or alphabet size

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-19
SLIDE 19

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Pattern Avoidance in Words Previous Work

Previous Work for Words

Results by...

Burstein: initial results, generating functions Albert, Aldred, Atkinson, Handley, Holton: results for specific 3-letter patterns Brändén, Mansour: automata for enumeration, for specific k

Note: most work is for specific patterns, would like a universal technique that works well regardless of pattern or alphabet size For permutations, one universal technique is Zeilberger and Vatter’s Enumeration Schemes.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-20
SLIDE 20

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Outline

1

Introduction/History Pattern Avoidance in Words Previous Work

2

Prefix Schemes for Words Definitions Examples Success Rate

3

Other Schemes for Words Schemes for Monotone Patterns

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-21
SLIDE 21

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Refinement

Main Idea: Can’t always directly find a recurrence to count A[a1,...,ak]({Q}) Instead, divide and conquer according to pattern formed by first i letters Look for recurrences between these subsets of A[a1,...,ak]({Q})

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-22
SLIDE 22

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Notation

When Q is understood, A[a1,...,ak]

  • p1 . . . pl
  • :=

{w ∈ [k]

ai | w has prefix p1 . . . pl}

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-23
SLIDE 23

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Notation

When Q is understood, A[a1,...,ak]

  • p1 . . . pl
  • :=

{w ∈ [k]

ai | w has prefix p1 . . . pl}

and, for 1 ≤ i1 ≤ · · · ≤ il ≤ k, A[a1,...,ak] p1 . . . pl i1 . . . il

  • :={w ∈ [k]

ai |

w has prefix p1 . . . pl and i1, . . . il are the first l letters of w}

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-24
SLIDE 24

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Refinement Example

We have, A[a1,...,ak]() = A[a1,...,ak](1) = A[a1,...,ak](12) ∪ A[a1,...,ak](11) ∪ A[a1,...,ak](21)

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-25
SLIDE 25

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Refinement Example

We have, A[a1,...,ak]() = A[a1,...,ak](1) = A[a1,...,ak](12) ∪ A[a1,...,ak](11) ∪ A[a1,...,ak](21)

  • r graphically:

∅ 1 12 11 21

  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-26
SLIDE 26

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Refinement

Main consideration: for permutations, only permutations appear as prefixes e.g. refinements of 1 are 12 and 21

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-27
SLIDE 27

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Refinement

Main consideration: for permutations, only permutations appear as prefixes e.g. refinements of 1 are 12 and 21 for words, there are many more prefixes e.g. refinements of 1 are 12, 21, and 11

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-28
SLIDE 28

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Reversibly Deletable

Given a prefix p = p1...pt, position r is reversibly deletable if every possible bad pattern involving pr implies another bad pattern without pr.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-29
SLIDE 29

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Reversibly Deletable

Given a prefix p = p1...pt, position r is reversibly deletable if every possible bad pattern involving pr implies another bad pattern without pr. For example, avoid q = 123, and let p = 21...

21...a...b

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-30
SLIDE 30

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Reversibly Deletable

Given a prefix p = p1...pt, position r is reversibly deletable if every possible bad pattern involving pr implies another bad pattern without pr. For example, avoid q = 123, and let p = 21...

21...a...b 21...a...b

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-31
SLIDE 31

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Reversibly Deletable

Given a prefix p = p1...pt, position r is reversibly deletable if every possible bad pattern involving pr implies another bad pattern without pr. For example, avoid q = 123, and let p = 21...

21...a...b 21...a...b

p1 = 2 is reversibly deletable for q = 123, p = 21.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-32
SLIDE 32

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Reversibly Deletable

There is always a natural embedding A[a1,...,an] p1 . . . pl i1 . . . il

  • → A[a1,...aj−1,...an]

p1 . . . ˆ pr . . . pl i1 . . .ˆ j . . . il

  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-33
SLIDE 33

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Reversibly Deletable

There is always a natural embedding A[a1,...,an] p1 . . . pl i1 . . . il

  • → A[a1,...aj−1,...an]

p1 . . . ˆ pr . . . pl i1 . . .ˆ j . . . il

  • If pr is reversibly deletable, and the role of pr is played by

letter j, then |A[a1,...,an] p1 . . . pl i1 . . . il

  • | = |A[a1,...aj−1,...an]

p1 . . . ˆ pr . . . pl i1 . . .ˆ j . . . il

  • |.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-34
SLIDE 34

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Reversibly Deletable Example

For Q = {123}, we have, A[a1,...,ak] 21 ij

  • = A[a1,...,aj−1,...ak]

1 i

  • A[a1,...,ak]

11 ij

  • = A[a1,...,ai−1,...ak]

1 j

  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-35
SLIDE 35

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Reversibly Deletable Example

For Q = {123}, we have, A[a1,...,ak] 21 ij

  • = A[a1,...,aj−1,...ak]

1 i

  • A[a1,...,ak]

11 ij

  • = A[a1,...,ai−1,...ak]

1 j

  • r graphically:

∅ 1 12 11 21

  • d1
  • d1
  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-36
SLIDE 36

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Reversibly Deletable

Main consideration: for permutations, reversibly deletable letters can always be removed together

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-37
SLIDE 37

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Reversibly Deletable

Main consideration: for permutations, reversibly deletable letters can always be removed together for words, two letters can be reversibly deletable separately but not together e.g. q = 123, p = 11

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-38
SLIDE 38

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Gap Vectors

Consider words that avoid q = 123 and begin with prefix p = 12 sorted prefix: 1 2 letters involved in prefix: i j vector: <a, b, c>

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-39
SLIDE 39

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Gap Vectors

Consider words that avoid q = 123 and begin with prefix p = 12 sorted prefix: 1 2 letters involved in prefix: i j vector: <a, b, c> sorted word: · · ·

  • ≥a

i · · ·

  • ≥b

j · · ·

  • ≥c

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-40
SLIDE 40

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Gap Vectors

Consider words that avoid q = 123 and begin with prefix p = 12 sorted prefix: 1 2 letters involved in prefix: i j vector: <a, b, c> sorted word: · · ·

  • ≥a

i · · ·

  • ≥b

j · · ·

  • ≥c

v is a gap vector for p if there are no words avoiding q with prefix p and spacing v.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-41
SLIDE 41

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Gap Vectors

Consider words that avoid q = 123 and begin with prefix p = 12 sorted prefix: 1 2 letters involved in prefix: i j vector: <a, b, c> sorted word: · · ·

  • ≥a

i · · ·

  • ≥b

j · · ·

  • ≥c

v is a gap vector for p if there are no words avoiding q with prefix p and spacing v. e.g. v =< 0, 0, 1 > is a gap vector for q = 123, p = 12.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-42
SLIDE 42

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Gap Vector Example

For Q = {123}, we have, A[a1,...,ak] 12 ij

  • = A[a1,...,ak]

12 ik

  • = A[a1,...,ak−1]

1 i

  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-43
SLIDE 43

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Gap Vector Example

For Q = {123}, we have, A[a1,...,ak] 12 ij

  • = A[a1,...,ak]

12 ik

  • = A[a1,...,ak−1]

1 i

  • r graphically:

∅ 1 12 11 21 ≥ (0, 0, 1)

  • d1
  • d1
  • d2
  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-44
SLIDE 44

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Gap Vectors

Main consideration: for permutations, <a,b,c> means · · ·

  • ≥a

i · · ·

  • ≥b

j · · ·

  • ≥c

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-45
SLIDE 45

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Gap Vectors

Main consideration: for permutations, <a,b,c> means · · ·

  • ≥a

i · · ·

  • ≥b

j · · ·

  • ≥c

for words, <a,b+1,c> means · · ·

  • ≥a

i · · ·

  • ≥b

j · · ·

  • ≥c

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-46
SLIDE 46

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Enumeration Schemes

Refinements Reversibly Deletable Elements Gap vectors can all be found completely automatically, so we have an algorithm to compute an enumeration schemes for words.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-47
SLIDE 47

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Implementation

Maple package mVATTER has the following functions SchemeF: input: set of patterns, maximum scheme depth (also faster version with maximum gap weight as input)

  • utput: scheme

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-48
SLIDE 48

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Implementation

Maple package mVATTER has the following functions SchemeF: input: set of patterns, maximum scheme depth (also faster version with maximum gap weight as input)

  • utput: scheme

MiklosA, MiklosTot: input: scheme, alphabet

  • utput: number of words obeying input

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-49
SLIDE 49

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Implementation

Maple package mVATTER has the following functions SchemeF: input: set of patterns, maximum scheme depth (also faster version with maximum gap weight as input)

  • utput: scheme

MiklosA, MiklosTot: input: scheme, alphabet

  • utput: number of words obeying input

SipurF: input: list of pattern lengths, max scheme depth

  • utput: scheme and statistics for every

equivalence class of patterns with lengths in list

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-50
SLIDE 50

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Outline

1

Introduction/History Pattern Avoidance in Words Previous Work

2

Prefix Schemes for Words Definitions Examples Success Rate

3

Other Schemes for Words Schemes for Monotone Patterns

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-51
SLIDE 51

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

The Simplest Examples

Av(∅) ∅ 1 Av(12) ∅ 1 ≥ (0, 1)

  • d1
  • d1
  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-52
SLIDE 52

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Isomorphic Prefix Schemes

Av(123) ∅ 1 12 ≥ (0, 1, 1) 11 21 Av(132) ∅ 1 12 ≥ (0, 2, 0) 11 21

  • d2
  • d1
  • d1
  • d2
  • d1
  • d1
  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-53
SLIDE 53

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Another Example

Av(1234) ∅ 1 12 11 21 132 123 ≥ (0, 1, 1, 1) 231 121 122 2312 2413 3412 3421 2311 2313 2314 ≥ (0, 1, 1, 1, 1)

  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-54
SLIDE 54

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Another Example

Av(1234) ∅ 1 12 11 21 132 123 ≥ (0, 1, 1, 1) 231 121 122 2312 2413 3412 3421 2311 2313 2314 ≥ (0, 1, 1, 1, 1)

  • d1
  • d1
  • d2
  • d3
  • d3
  • d2
  • d1,2
  • d1,2
  • d1,2
  • d3
  • d3
  • d4
  • d4
  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-55
SLIDE 55

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Outline

1

Introduction/History Pattern Avoidance in Words Previous Work

2

Prefix Schemes for Words Definitions Examples Success Rate

3

Other Schemes for Words Schemes for Monotone Patterns

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-56
SLIDE 56

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Statistics

success rate is bounded above by success rate for permutation schemes

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-57
SLIDE 57

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Statistics

success rate is bounded above by success rate for permutation schemes Pattern Lengths Permutations Words [2] 1/1 (100%) 1/1 (100%) [2,3] 1/1 (100%) 1/1 (100%) [2,4] 1/1 (100%) 1/1 (100%) [3] 2/2 (100%) 2/2 (100%) [3,3] 5/5 (100%) 6/6 (100%) [3,3,3] 5/5 (100%) 6/6 (100%) [3,3,3,3] 5/5 (100%) 6/6 (100%) [3,3,3,3,3] 2/2 (100%) 2/2 (100%)

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-58
SLIDE 58

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Statistics

Pattern Lengths Permutations Words [4] 2/7 (28.6%) 2/8 (25%) [3,4] 17/18 (94.4%) 9/24 (37.5%) [3,3,4] 23/23 (100%) 27/31 (87.1%) [3,3,3,4] 16/16 (100%) 20/20 (100%) [3,3,3,3,4] 6/6 (100%) 6/6 (100%) [3,3,3,3,3,4] 1/1 (100%) 1/1 (100%) [4,4] 29/56 (51.8%) ?/84 (in process) [3,4,4] 92/92 (100%) 38/146 (26%) [3,3,4,4] 68/68 (100%) 89/103 (86.4%) [3,3,3,4,4] 23/23 (100%) 29/29 (100%) [3,3,3,3,4,4] 3/3 (100%) 3/3 (100%)

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-59
SLIDE 59

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Avoiding a Pattern With Repeated Letters

  • nly works to avoid permutation patterns

Let q = q1lq2lq3, where l is the first repeated letter in q.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-60
SLIDE 60

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Avoiding a Pattern With Repeated Letters

  • nly works to avoid permutation patterns

Let q = q1lq2lq3, where l is the first repeated letter in q.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-61
SLIDE 61

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Definitions Examples Success Rate

Avoiding a Pattern With Repeated Letters

  • nly works to avoid permutation patterns

Let q = q1lq2lq3, where l is the first repeated letter in q.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-62
SLIDE 62

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

Another Direction

Zeilberger’s schemes: patterns formed by the first i letters

  • f words

(refinement by adding one letter at a time) drawback: only works for permutation-avoiding words

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-63
SLIDE 63

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

Another Direction

Zeilberger’s schemes: patterns formed by the first i letters

  • f words

(refinement by adding one letter at a time) drawback: only works for permutation-avoiding words Vatter’s schemes: patterns formed by the smallest i letters

  • f words

drawback: 1 → 11 → 111 → . . . is a problem

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-64
SLIDE 64

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

Another Direction

Zeilberger’s schemes: patterns formed by the first i letters

  • f words

(refinement by adding one letter at a time) drawback: only works for permutation-avoiding words Vatter’s schemes: patterns formed by the smallest i letters

  • f words

drawback: 1 → 11 → 111 → . . . is a problem Solution: following Vatter, consider the patterns formed by the smallest letters of words BUT refine by adding all copies of a letter simultaneously

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-65
SLIDE 65

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

Outline

1

Introduction/History Pattern Avoidance in Words Previous Work

2

Prefix Schemes for Words Definitions Examples Success Rate

3

Other Schemes for Words Schemes for Monotone Patterns

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-66
SLIDE 66

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

Av(112)

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-67
SLIDE 67

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

Av(112) ∅

  • ne 1

≥ 2 1s first repeated letter in position a

  • insert at end
  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-68
SLIDE 68

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

Av(112) ∅ 1 1 a

  • a=2
  • insert at end
  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-69
SLIDE 69

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

Av(112) ∅ 1 1 a

  • ne 2

1 still 1st repeated letter b ≥ 2 2s

  • a=2
  • insert at end
  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-70
SLIDE 70

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

Av(112) ∅ 1 1 a

  • ne 2

1 still 1st repeated letter (a − 1) old letters + one 2 1 a b ≥ 2 2s

  • a=2
  • insert at end
  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-71
SLIDE 71

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

Av(112) ∅ 1 1 a

  • (a − 1)old letters

+ one 2

  • 1

b ≥ 2 2s

  • a=2
  • insert at end
  • a∗

 

1 a + 1

 

  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-72
SLIDE 72

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

Av(112) ∅ 1 1 a

  • (a − 1)old letters

+ one 2

  • 1

b ≥ 2 2s (k − 2)old letters + one 2 2 k

  • a=2
  • insert at end
  • a∗

 

1 a + 1

 

  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-73
SLIDE 73

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

Av(112) ∅ 1 1 a

  • (a − 1)old letters

+ one 2

  • 1

b ≥ 2 2s (k − 2)old letters + one 2 2 k (a − 1 − (k − 2)) old letters + b-2 2s 1 a

  • a=2
  • insert at end
  • a∗

 

1 a + 1

 

  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-74
SLIDE 74

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

Av(112) ∅ 1 1 a

  • (a − 1) old letters

+ one 2

  • 1

(a − 1)1 old letters +b 2s

  • 1
  • a=2
  • insert at end
  • a∗

 

1 a + 1

 

  • a+1

k=2 (k−1 1 )((a−1)−(k−2) (b−2)

)

 1

k

 

  • Lara Pudwell

Schemes for Pattern-Avoiding Words

slide-75
SLIDE 75

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

Let A[a1,...,ak] := |{w ∈ [k]

ai|w has ai is, w avoids 112}|

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-76
SLIDE 76

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

Let A[a1,...,ak] := |{w ∈ [k]

ai|w has ai is, w avoids 112}|

B(i)

[a1,...,ak] := |{w ∈ A[a1,...,ak]|

w’s first repeated letter is in position i}|

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-77
SLIDE 77

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

We now have: A[a1,...,ak] =      1 k = 1 a2+···+ak+1

1

  • A[a2,...,ak]

k > 1, a1 = 1 B(2)

[a2,...,ak]

k > 1, a1 > 1

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-78
SLIDE 78

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

We now have: A[a1,...,ak] =      1 k = 1 a2+···+ak+1

1

  • A[a2,...,ak]

k > 1, a1 = 1 B(2)

[a2,...,ak]

k > 1, a1 > 1 B(i)

[a1,...,ak] =

       i−1+a1

a1

  • k = 1

i ∗ B(i+1)

[a2,...,ak]

a1 = 1 i+1

k=2(k − 1)

(i−1)−(k−2)+(a1−2)

a1−2

  • B(k)

[a2,...,ak]

a1 > 1

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-79
SLIDE 79

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

An Example

We now have: A[a1,...,ak] =      1 k = 1 a2+···+ak+1

1

  • A[a2,...,ak]

k > 1, a1 = 1 B(2)

[a2,...,ak]

k > 1, a1 > 1 B(i)

[a1,...,ak] =

       i−1+a1

a1

  • k = 1

i ∗ B(i+1)

[a2,...,ak]

a1 = 1 i+1

k=2(k − 1)

(i−1)−(k−2)+(a1−2)

a1−2

  • B(k)

[a2,...,ak]

a1 > 1 A[a1,...,ak] =

k

  • i=2

(aj + · · · ak + 1) .

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-80
SLIDE 80

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

Non-prefix schemes

This example can be generalized to find a scheme for words avoiding any monotone pattern.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-81
SLIDE 81

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Monotone Patterns

Non-prefix schemes

This example can be generalized to find a scheme for words avoiding any monotone pattern. Currently exploring extensions to other types of patterns.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-82
SLIDE 82

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary

Summary

There are few techniques to count large classes of pattern-avoiding words.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-83
SLIDE 83

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary

Summary

There are few techniques to count large classes of pattern-avoiding words. Extending Zeilberger’s and Vatter’s schemes gives a good success rate for words avoiding permutations and for words avoiding monotone patterns.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-84
SLIDE 84

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary

Summary

There are few techniques to count large classes of pattern-avoiding words. Extending Zeilberger’s and Vatter’s schemes gives a good success rate for words avoiding permutations and for words avoiding monotone patterns. Future work

Find other general techniques for enumerating classes of permutation-avoiding words.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-85
SLIDE 85

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary

Summary

There are few techniques to count large classes of pattern-avoiding words. Extending Zeilberger’s and Vatter’s schemes gives a good success rate for words avoiding permutations and for words avoiding monotone patterns. Future work

Find other general techniques for enumerating classes of permutation-avoiding words. Simplify schemes to compute more data more quickly.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-86
SLIDE 86

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary

Summary

There are few techniques to count large classes of pattern-avoiding words. Extending Zeilberger’s and Vatter’s schemes gives a good success rate for words avoiding permutations and for words avoiding monotone patterns. Future work

Find other general techniques for enumerating classes of permutation-avoiding words. Simplify schemes to compute more data more quickly. Convert concrete enumeration schemes to closed forms.

Lara Pudwell Schemes for Pattern-Avoiding Words

slide-87
SLIDE 87

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary

References

  • M. Albert, R. Aldred, M.D. Atkinson, C. Handley, D. Holton,

Permutations of a multiset avoiding permutations of length 3, Europ. J.

  • Combin. 22, 1021-1031 (2001).

P . Branden, T. Mansour, Finite automata and pattern avoidance in words, Joural Combinatorial Theory Series A 110:1, 127-145 (2005). Alexander Burstein, Enumeration of Words with Forbidden Patterns, Ph.D. Thesis, University of Pennsylvania, 1998. Vince Vatter, Enumeration Schemes for Restricted Permutations, Combinatorics, Probability, and Computing, to appear. Doron Zeilberger, Enumeration Schemes, and More Importantly, Their Automatic Generation, Annals of Combinatorics 2, 185-195 (1998). Doron Zeilberger, On Vince Vatter’s Brilliant Extension of Doron Zeilberger’s Enumeration Schemes for Herb Wilf’s Classes, The Personal Journal of Ekhad and Zeilberger, 2006. http://www.math.rutgers.edu/∼ zeilberg/pj.html.

Lara Pudwell Schemes for Pattern-Avoiding Words