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Pattern-Avoiding Permutations Enumeration Schemes Summary An Introduction to Enumeration Schemes Lara Pudwell Valparaiso University Trinity University Mathematics Colloquium November 18, 2009 Lara Pudwell An Introduction to Enumeration

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SLIDE 1

Pattern-Avoiding Permutations Enumeration Schemes Summary

An Introduction to Enumeration Schemes

Lara Pudwell

Valparaiso University

Trinity University Mathematics Colloquium November 18, 2009

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 2

Pattern-Avoiding Permutations Enumeration Schemes Summary

Outline

1

Pattern-Avoiding Permutations Definitions Counting Results Motivation

2

Enumeration Schemes Divide Conquer Putting It All Together...

3

Summary

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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 An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Pattern Avoidance/Containment Given permutations π = π1 · · · πn and q = q1 · · · qm, π contains q as a pattern if there is 1 ≤ i1 < · · · < im ≤ n so that πi1 · · · πim reduces to q;

  • therwise π avoids q.

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

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Two Questions Easy: Given π ∈ Sn, what patterns does π contain? Hard: Given q ∈ Sm, Let Sn(q) = {π ∈ Sn | π avoids q}. Find an expression for |Sn(q)|.

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 8

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 9

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 10

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 11

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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 An Introduction to Enumeration Schemes

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SLIDE 12

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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 An Introduction to Enumeration Schemes

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SLIDE 13

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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 An Introduction to Enumeration Schemes

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SLIDE 14

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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 An Introduction to Enumeration Schemes

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SLIDE 15

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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 An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding a Pattern of Length 3 There are six patterns of length 3: 123, 132, 213, 231, 312, 321. Using Wilf equivalence, we have |Sn(123)| = |Sn(321)| and |Sn(132)| = |Sn(231)| = |Sn(213)| = |Sn(312)|.

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 18

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding a Pattern of Length 3 There are six patterns of length 3: 123, 132, 213, 231, 312, 321. Using Wilf equivalence, 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 An Introduction to Enumeration Schemes

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SLIDE 19

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding the pattern 132 What is |Sn(132)|?

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 20

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding the pattern 132 What is |Sn(132)|?

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 21

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding the pattern 132 What is |Sn(132)|?

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 22

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding the pattern 132 What is |Sn(132)|?

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 23

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding the pattern 132 What is |Sn(132)|?

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 24

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding the pattern 132 What is |Sn(132)|?

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 25

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding the pattern 132 What is |Sn(132)|?

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 26

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding the pattern 132 What is |Sn(132)|? |Sn(132)| =

n

  • i=1

|Si−1(132)| · |Sn−i(132)| (for n > 0)

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 27

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

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 An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding a Pattern of Length 4 There are 24 patterns of length 4. Using Wilf equivalence and similar bijections, we can narrow

  • ur work to 3 cases:

Sn(1342), Sn(1234), and Sn(1324).

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 29

Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Avoiding a Pattern of Length 4 There are 24 patterns of length 4. Using Wilf equivalence and similar bijections, we can narrow

  • ur 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 An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Pattern-Avoidance Sightings Pattern-avoiding permutations appear in the context of... sorting algorithms Schubert varieties experimental mathematics

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Counting Results Motivation

Algorithms for Pattern-Avoiding Permutations Most techniques studying |Sn(q)| find formulas for a specific q. 1998: Zeilberger’s prefix enumeration schemes, i.e. a system of recurrences to compute |Sn(q)|. 2005: Vatter’s modified schemes automate the computation of |Sn(q)| for even more patterns q.

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Refinement Notation Goal: Divide Sn(q) into disjoint subsets. Sn

  • q; p1 · · · pl
  • :=
  • π ∈ Sn
  • π avoids q

π has prefix p1 · · · pl

  • For example, S3(123) = {132, 213, 231, 312, 321}, so

S3(123; 12) = {132, 231}, and S3(123, 21) = {213, 312, 321}.

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Refinement Notation Goal: Divide Sn(q) into subsets. Sn

  • q; p1 · · · pl

i1 · · · il

  • :=

  π ∈ Sn

  • π avoids q

π has prefix p1 · · · pl π = i1 · · · ilπl+1 · · · πn    We have seen S3(123; 12) = {132, 231}, so S3

  • 123, 12

13

  • = {132},

S3

  • 123, 12

23

  • = {231}, and

S3

  • 123, 12

12

  • = {}.

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Refinement Given a prefix p of length l, the refinements of p (Ref(p)) are the permutations of length l + 1 whose first l letters reduce to p. For example, Ref(231) = {3421, 3412, 2413, 2314}. We have Sn(q; p) =

  • r∈Ref(p)

Sn(q; r)

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Refinement Given a prefix p of length l, the refinements of p (Ref(p)) are the permutations of length l + 1 whose first l letters reduce to p. For example, Ref(231) = {3421, 3412, 2413, 2314}. We have Sn(q; p) =

  • r∈Ref(p)

Sn(q; r)

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 36

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Refinement Given a prefix p of length l, the refinements of p (Ref(p)) are the permutations of length l + 1 whose first l letters reduce to p. For example, Ref(231) = {3421, 3412, 2413, 2314}. We have Sn(q; p) =

  • r∈Ref(p)

Sn(q; r)

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 37

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Refinement Given a prefix p of length l, the refinements of p (Ref(p)) are the permutations of length l + 1 whose first l letters reduce to p. For example, Ref(231) = {3421, 3412, 2413, 2314}. We have Sn(q; p) =

  • r∈Ref(p)

Sn(q; r)

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 38

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Refinement Given a prefix p of length l, the refinements of p (Ref(p)) are the permutations of length l + 1 whose first l letters reduce to p. For example, Ref(231) = {3421, 3412, 2413, 2314}. We have Sn(q; p) =

  • r∈Ref(p)

Sn(q; r)

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Refinement Example For any pattern q, Sn(q) = Sn(q; 1) = Sn(q; 12) ∪ Sn(q; 21) = . . . .

  • r graphically:

∅ 1 12 21

  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Reversibly Deletable Positions Reversibly Deletable Positions Given a pattern q and a prefix p, pr is reversibly deletable if Deleting pr from π ∈ Sn(q; p1 · · · pl) produces a q-avoiding permutation of length n − 1, and Inserting pr into π ∈ Sn−1(q; p1 · · · pr−1pr+1 · · · pl) produces a q-avoiding permutation of length n. If pr is reversibly deletable then, |Sn(q; p1 · · · pl)| = |Sn−1(q; p1 · · · pr−1pr+1 · · · pl)| .

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Reversibly Deletable Positions For 123-avoiding permutations that begin with p = 21, p1 is reversibly deletable

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Reversibly Deletable Positions For 123-avoiding permutations that begin with p = 21, p1 is reversibly deletable

  • Sn
  • 123; 21

ij

  • =
  • Sn−1
  • 123; 1

j

  • Lara Pudwell

An Introduction to Enumeration Schemes

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SLIDE 43

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Reversibly Deletable Positions For 123-avoiding permutations that begin with p = 21, p2 is not reversibly deletable.

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Reversibly Deletable Positions For 123-avoiding permutations that begin with p = 21, p2 is not reversibly deletable. While deleting p2 gives a smaller 123-avoiding permutation, inserting p2 into a member of Sn−1(123) doesn’t always give a 123-avoiding permutation.

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Algorithm to find Reversibly Deletable Elements Brute force: List all scenarios in which pr can participate in a forbidden q-pattern. Delete pr from each scenario. If every resulting permutation contains q, then pr is reversibly deletable. In practice: Theorem (Vatter, 2005) If |Sn(q; p1 · · · pl)| = |Sn−1(q; p1 · · · pr−1pr+1 · · · pl)| for all n ≤ |p| + |q| − 1 then pr is reversibly deletable.

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Reversibly Deletable Example Graphically, for q = 123, we have: ∅ 1 12 21

  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Reversibly Deletable Example Graphically, for q = 123, we have: ∅ 1 12 21

  • d1
  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Gap Vectors Gap vectors give a condition for which choices of i1, . . . , il yield

  • Sn
  • q; p1 · · · pr · · · pl

i1 · · · ir · · · il

  • = 0.

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 49

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Gap Vectors Gap vectors give a condition for which choices of i1, . . . , il yield

  • Sn
  • q; p1 · · · pr · · · pl

i1 · · · ir · · · il

  • = 0.

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Gap Vectors Gap vectors give a condition for which choices of i1, . . . , il yield

  • Sn
  • q; p1 · · · pr · · · pl

i1 · · · ir · · · il

  • = 0.

Since there are no members of Sn(123; 12) where v3 = 1, we say (0, 0, 1) is a gap vector for p = 12

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 51

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Gap Vectors Knowing that (0, 0, 1) is a gap vector for q = 123 and p = 12 can help us determine more reversibly deletable positions.

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 52

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Gap Vectors Knowing that (0, 0, 1) is a gap vector for q = 123 and p = 12 can help us determine more reversibly deletable positions.

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 53

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Gap Vectors Knowing that (0, 0, 1) is a gap vector for q = 123 and p = 12 can help us determine more reversibly deletable positions.

  • Sn
  • 123; 12

ij

  • = 0 if j < n
  • Sn
  • 123; 12

in

  • =
  • Sn−1
  • 123; 1

i

  • Lara Pudwell

An Introduction to Enumeration Schemes

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SLIDE 54

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Algorithm to find Gap Vectors Brute force: List all permutations π that begin with prefix p and obey vector v. If every element of this set contains q, then v is a gap vector. In practice: Theorem (Vatter, 2005)

1

If v is a gap vector for (q; p), and u ≥ v componentwise, then u is a gap vector for (q; p).

2

Minimal gap vectors for (q; p) have ||v|| ≤ |q| − 1.

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Gap Vectors Graphically, for q = 123, we have: ∅ 1 12 21

  • d1
  • Lara Pudwell

An Introduction to Enumeration Schemes

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SLIDE 56

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Gap Vectors Graphically, for q = 123, we have: ∅ 1 12 21 ≥ (0, 0, 1)

  • d1
  • Lara Pudwell

An Introduction to Enumeration Schemes

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SLIDE 57

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Gap Vectors Graphically, for q = 123, we have: ∅ 1 12 21 ≥ (0, 0, 1)

  • d1
  • d2
  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Definition An enumeration scheme is a set of triples [pi, Gi, Ri] such that for each triple pi is a reduced prefix of length n Gi is a set of vectors of length n + 1 Ri a subset of {1, . . . , n} and either Ri is non-empty or all refinements of pi are also in the scheme.

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Definition An enumeration scheme is a set of triples [pi, Gi, Ri] such that for each triple pi is a reduced prefix of length n (prefix) Gi is a set of vectors of length n + 1 (gap vectors) Ri a subset of {1, . . . , n} (reversibly deletable positions) and either Ri is non-empty or all refinements of pi are also in the scheme.

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example For the pattern q = 123, we have constructed the following scheme: S = {[∅, ∅, ∅]} ∅

Lara Pudwell An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example For the pattern q = 123, we have constructed the following scheme: S = {[∅, ∅, ∅], [1, ∅, ∅]} ∅ 1

  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example For the pattern q = 123, we have constructed the following scheme: S = {[∅, ∅, ∅], [1, ∅, ∅], [12, G12, R12], [21, G21, R21]} ∅ 1

  • 12

21

  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example For the pattern q = 123, we have constructed the following scheme: S = {[∅, ∅, ∅], [1, ∅, ∅], [12, G12, R12], [21, ∅, {1}]} ∅ 1

  • 12

21

  • d1
  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example For the pattern q = 123, we have constructed the following scheme: S = {[∅, ∅, ∅], [1, ∅, ∅], [12, {(0, 0, 1)}, R12], [21, ∅, {1}]} ∅ 1

  • 12

21 ≥ (0, 0, 1)

  • d1
  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example For the pattern q = 123, we have constructed the following scheme: S = {[∅, ∅, ∅], [1, ∅, ∅], [12, {(0, 0, 1)}, {2}], [21, ∅, {1}]} ∅ 1 12 21 ≥ (0, 0, 1)

  • d1
  • d2
  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example For the pattern q = 123, we have constructed the following scheme: S = {[∅, ∅, ∅], [1, ∅, ∅], [12, {(0, 0, 1)}, {2}], [21, ∅, {1}]} ∅ 1 12 21 ≥ (0, 0, 1)

  • d1
  • d2
  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example The scheme can S = {[∅, ∅, ∅], [1, ∅, ∅], [12, {(0, 0, 1)}, {2}], [21, ∅, {1}]} can be seen as a recurrence to count the elements of Sn(123). |Sn(123)| = n

i=1

  • Sn
  • 123, 1

i

  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example The scheme can S = {[∅, ∅, ∅], [1, ∅, ∅], [12, {(0, 0, 1)}, {2}], [21, ∅, {1}]} can be seen as a recurrence to count the elements of Sn(123). |Sn(123)| = n

i=1

  • Sn
  • 123, 1

i

  • = n

i=1

n

j=i+1

  • Sn
  • 123; 12

ij

  • + n

i=1

i−1

h=1

  • Sn
  • 123; 21

ih

  • Lara Pudwell

An Introduction to Enumeration Schemes

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Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example The scheme can S = {[∅, ∅, ∅], [1, ∅, ∅], [12, {(0, 0, 1)}, {2}], [21, ∅, {1}]} can be seen as a recurrence to count the elements of Sn(123). |Sn(123)| = n

i=1

  • Sn
  • 123, 1

i

  • = n

i=1

n

j=i+1

  • Sn
  • 123; 12

ij

  • + n

i=1

i−1

h=1

  • Sn
  • 123; 21

ih

  • = n

i=1

  • Sn
  • 123; 12

in

  • + n

i=1

i−1

h=1

  • Sn−1
  • 123; 1

h

  • Lara Pudwell

An Introduction to Enumeration Schemes

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SLIDE 70

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example The scheme can S = {[∅, ∅, ∅], [1, ∅, ∅], [12, {(0, 0, 1)}, {2}], [21, ∅, {1}]} can be seen as a recurrence to count the elements of Sn(123). |Sn(123)| = n

i=1

  • Sn
  • 123, 1

i

  • = n

i=1

n

j=i+1

  • Sn
  • 123; 12

ij

  • + n

i=1

i−1

h=1

  • Sn
  • 123; 21

ih

  • = n

i=1

  • Sn
  • 123; 12

in

  • + n

i=1

i−1

h=1

  • Sn−1
  • 123; 1

h

  • = n

i=1

  • Sn−1
  • 123; 1

i

  • + i−1

h=1

  • Sn−1
  • 123; 1

h

  • Lara Pudwell

An Introduction to Enumeration Schemes

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SLIDE 71

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Scheme Example The scheme can S = {[∅, ∅, ∅], [1, ∅, ∅], [12, {(0, 0, 1)}, {2}], [21, ∅, {1}]} can be seen as a recurrence to count the elements of Sn(123). |Sn(123)| = n

i=1

  • Sn
  • 123, 1

i

  • = n

i=1

n

j=i+1

  • Sn
  • 123; 12

ij

  • + n

i=1

i−1

h=1

  • Sn
  • 123; 21

ih

  • = n

i=1

  • Sn
  • 123; 12

in

  • + n

i=1

i−1

h=1

  • Sn−1
  • 123; 1

h

  • = n

i=1

  • Sn−1
  • 123; 1

i

  • + i−1

h=1

  • Sn−1
  • 123; 1

h

  • = n

i=1

i

h=1

  • Sn−1
  • 123; 1

h

  • Lara Pudwell

An Introduction to Enumeration Schemes

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SLIDE 72

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Enumeration Schemes Refinements Reversibly deletable elements Gap vectors can all be found completely automatically, so we have an algorithm to compute enumeration schemes for pattern-avoiding permutations.

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 73

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Sn(∅) and Sn(12) Sn(∅) ∅ 1 Sn(12) ∅ 1 ≥ (0, 1)

  • d1
  • d1
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An Introduction to Enumeration Schemes

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SLIDE 74

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Sn(∅) and Sn(12) Sn(∅) ∅ 1 Sn(12) ∅ 1 ≥ (0, 1)

  • d1
  • d1
  • |Sn(∅)| =

n

  • i=1
  • Sn
  • ∅; 1

i

  • =

n

  • i=1

|Sn−1 (∅)| = n |Sn−1(∅)|

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 75

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Sn(∅) and Sn(12) Sn(∅) ∅ 1 Sn(12) ∅ 1 ≥ (0, 1)

  • d1
  • d1
  • |Sn(12)| =

n

  • i=1
  • Sn
  • 12; 1

i

  • =
  • Sn
  • 12; 1

n

  • = |Sn−1 (12)|

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 76

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Sn(123) and Sn(132) Sn(123) ∅ 1 12 ≥ (0, 0, 1) 21 Sn(132) ∅ 1 12 ≥ (0, 1, 0) 21

  • d2
  • d1
  • d2
  • d1
  • Lara Pudwell

An Introduction to Enumeration Schemes

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SLIDE 77

Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Conquer Putting It All Together...

Sn(1234) Sn(1234) ∅ 1 12 21 132 123 ≥ (0, 0, 0, 1) 231 2413 3412 3421 2314 ≥ (0, 0, 0, 0, 1)

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

An Introduction to Enumeration Schemes

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SLIDE 78

Pattern-Avoiding Permutations Enumeration Schemes Summary Summary

Summary There are few techniques to count many classes of pattern-avoiding permutations. Zeilberger’s and Vatter’s schemes give a good success rate for counting the elements of Sn(q). Enumeration schemes have also been successfully used to count:

pattern-avoiding words (strings with repeated letters) permutations avoiding barred patterns

Lara Pudwell An Introduction to Enumeration Schemes

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SLIDE 79

Pattern-Avoiding Permutations Enumeration Schemes Summary Summary

Thank You!

Lara Pudwell An Introduction to Enumeration Schemes