Classical pattern distribution in S n (132) and S n (123) Dun Qiu - - PowerPoint PPT Presentation

Classical pattern distribution in Sn(132) and Sn(123)

Dun Qiu UC San Diego duqiu@ucsd.edu Based on joint work with Jeffrey Remmel Permutation Patterns 2018, Dartmouth College July 11, 2018

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 1 / 28

In Memory of Jeffrey Remmel

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 2 / 28

Outline

1

Motivation

2

Introduction

3

Wilf-equivalence of Qγ

λ(t, x) 4

Recursions of Qγ

λ(t, x) 5

Other Results and Open Problems

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 3 / 28

Outline

1

Motivation

2

Introduction

3

Wilf-equivalence of Qγ

λ(t, x) 4

Recursions of Qγ

λ(t, x) 5

Other Results and Open Problems

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 3 / 28

Motivation

Ran Pan’s Project P http://www.math.ucsd.edu/∼projectp/ Problem 13: enumerate permutations in Sn avoiding a classical pattern and a consecutive pattern at the same time. Then Professor Remmel conducted researchs on distribution of classical patterns and consecutive patterns in Sn(132) and Sn(123).

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 4 / 28

Outline

1

Motivation

2

Introduction

3

Wilf-equivalence of Qγ

λ(t, x) 4

Recursions of Qγ

λ(t, x) 5

Other Results and Open Problems

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 4 / 28

Permutations, Descents, LRmins

A permutation σ = σ1 · · · σn of [n] = {1, . . . , n} is a rearrangement of the numbers 1, . . . , n. The set of permutations of [n] is denoted by Sn. σi is a descent if σi > σi+1. des(σ) is the number of descents in σ. We let LRmin(σ) denote the number of left to right minima of σ.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 5 / 28

Inversions, Coinversions

(σi, σj) is an inversion if i < j and σi > σj. inv(σ) denotes the number of inversions in σ. (σi, σj) is a coinversion if i < j and σi < σj. coinv(σ) denotes the number of coinversions in σ.

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Reduction of A Sequence

Given a sequence of distinct positive integers w = w1 . . . wn, we let the reduction (or standardization) of the sequence, red(w), denote the permutation of [n] obtained from w by replacing the i-th smallest letter in w by i.

Example

If w = 4592, then red(w) = 2341.

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Classical Patterns Occurrence and Avoidance

Given a permutation τ = τ1 . . . τj in Sj, we say the pattern τ occurs in σ = σ1 . . . σn ∈ Sn if there exist 1 ≤ i1 < · · · < ij ≤ n such that red(σi1 . . . σij) = τ. We let occrτ(σ) denote the number of τ occurrence in σ. We say σ avoids the pattern τ if τ does not occur in σ.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 8 / 28

Classical Patterns Occurrence and Avoidance

Given a permutation τ = τ1 . . . τj in Sj, we say the pattern τ occurs in σ = σ1 . . . σn ∈ Sn if there exist 1 ≤ i1 < · · · < ij ≤ n such that red(σi1 . . . σij) = τ. We let occrτ(σ) denote the number of τ occurrence in σ. We say σ avoids the pattern τ if τ does not occur in σ.

Example

π = 867932451 avoids pattern 132, contains pattern 123. occr123(π) = 2 since pattern occurrences are 6, 7, 9 and 3, 4, 5.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 8 / 28

Classical Patterns Occurrence and Avoidance

Given a permutation τ = τ1 . . . τj in Sj, we say the pattern τ occurs in σ = σ1 . . . σn ∈ Sn if there exist 1 ≤ i1 < · · · < ij ≤ n such that red(σi1 . . . σij) = τ. We let occrτ(σ) denote the number of τ occurrence in σ. We say σ avoids the pattern τ if τ does not occur in σ.

Example

π = 867932451 avoids pattern 132, contains pattern 123. occr123(π) = 2 since pattern occurrences are 6, 7, 9 and 3, 4, 5. τ is called a classical pattern. inversion − → pattern 21, coinversion − → pattern 12.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 8 / 28

Sn(σ)

We let Sn(λ) denote the set of permutations in Sn avoiding λ.

• Sn(132)
• =
• Sn(123)
• = Cn =

1 n+1

2n

n

• , the nth Catalan number.

Cn is also the number of n × n Dyck paths. Let Λ = {λ1, . . . , λr}, then Sn(Λ) is the set of permutations in Sn avoiding λ1, . . . , λr.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 9 / 28

Our Problem

Given two sets of permutations Λ = {λ1, . . . , λr} and Γ = {γ1, . . . , γs}, we study the distribution of classical patterns γ1, . . . , γs in Sn(Λ). Especially, we study pattern τ distribution in Sn(132) and Sn(123) in the case when τ is of length 3 and some special form.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 10 / 28

Generating Function

We define QΓ

Λ(t, x1, . . . , xs) = 1 +

• n≥1

tnQΓ

n,Λ(x1, . . . , xs),

where QΓ

n,Λ(x1, . . . , xs) =

• σ∈Sn(Λ)

x

• ccrγ1(σ)

1

· · · xoccrγs (σ)

s

. Especially, we have Qγ

λ(t, x) = 1 +

• n≥1

tnQγ

n,λ(x) and Qγ n,λ(x) =

• σ∈Sn(λ)

xoccrγ(σ).

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 11 / 28

Outline

1

Motivation

2

Introduction

3

Wilf-equivalence of Qγ

λ(t, x) 4

Recursions of Qγ

λ(t, x) 5

Other Results and Open Problems

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 11 / 28

Wilf-equivalence

Given a permutation σ, we denote the reverse of σ by σr, the complement

• f σ by σc, the reverse-complement of σ by σrc, and the inverse of σ by

σ−1.

Example

Let σ = 15324, then σr = 42351, σc = 51342, σrc = 24315, σ−1 = 14352.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 12 / 28

Wilf-equivalence

Sn(123) is closed under the operation reverse-complement. Both Sn(123) and Sn(132) are closed under the operation inverse. Thus,

Theorem

Given any permutation pattern γ, Qγ

123(t, x) = Qγrc 123(t, x) = Qγ−1 123 (t, x),

Qγ

132(t, x) = Qγ−1 132 (t, x).

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 13 / 28

Wilf-equivalence

When we let γ be a pattern of length 3,

Corollary

There are 4 Wilf-equivalent classes for Sn(132), (1) Q123

132(t, x),

(2) Q213

132(t, x),

(3) Q231

132(t, x) = Q312 132(t, x),

(4) Q321

132(t, x),

and there are 3 Wilf-equivalent classes for Sn(123), (1) Q132

123(t, x) = Q213 123(t, x),

(2) Q231

123(t, x) = Q312 123(t, x),

(3) Q321

123(t, x).

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 14 / 28

Outline

1

Motivation

2

Introduction

3

Wilf-equivalence of Qγ

λ(t, x) 4

Recursions of Qγ

λ(t, x) 5

Other Results and Open Problems

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 14 / 28

Method – Using Dyck Path Bijections

We use Dyck path bijections to calculate the recursive formulas for Qγ

λ(t, x).

Krattenthaler Φ : Sn(132) → Dn, Elizalde and Deutsch Ψ : Sn(123) → Dn.

8 6 7 9 4 3 2 5 1 8 6 9 7 4 3 2 5 1

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 15 / 28

Method – Using Dyck Path Bijections

Then, we the recursion of Dyck path by breaking the path at the first place it hits the diagonal to break it into 2 Dyck paths. Let D(x) be the generating function enumerating the number of Dyck paths of size n, D(x) = 1 + xD(x)2.

D(x) D(x) x

Recursion of Dyck path

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 16 / 28

Counting Length 2 pattern in Sn(132)

We first consider permutations that are avoiding 132 and the distribution

• f pattern of length 2, i.e. inv and coinv.

We let Qn(q) = Q12

n,132(q) =

• σ∈Sn(132)

qcoinv(σ), Q(t, q) = Q12

132(t, q) = 1 +

• n≥1

tn

• σ∈Sn(132)

qcoinv(σ), and Pn(p, q) =

• σ∈Sn(132)

pinv(σ)qcoinv(σ).

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 17 / 28

Counting Length 2 pattern in Sn(132)

We first consider permutations that are avoiding 132 and the distribution

• f pattern of length 2, i.e. inv and coinv.

We let Qn(q) = Q12

n,132(q) =

• σ∈Sn(132)

qcoinv(σ), Q(t, q) = Q12

132(t, q) = 1 +

• n≥1

tn

• σ∈Sn(132)

qcoinv(σ), and Pn(p, q) =

• σ∈Sn(132)

pinv(σ)qcoinv(σ). Since inv(σ) + coinv(σ) = n

2

• , we have the following relation about

Pn(p, q) and Qn(q), Pn(p, q) =

• σ∈Sn(132)

p(n

2)−coinv(σ)qcoinv(σ) = p(n 2)Qn

q p

• .

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 17 / 28

Counting Length 2 pattern in Sn(132)

Qn(q) – q-Catalan number.

Theorem (F¨ urlinger and Hofbauer)

Let Qn(q) = Q12

n,132(q) and Q(t, q) = Q12 132(t, q), then we have the

recursions, Q0(q) = 1, Qn(q) =

n

• k=1

qk−1Qk−1(q)Qn−k(q), (1) P0(q) = 1, Pn(q) =

n

• k=1

qk(n−k)Pk−1(q)Pn−k(q), (2) and we have the functional equation, Q(t, q) = 1 + tQ(t, q) · Q(tq, q). (3)

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 18 / 28

Counting Length 3 pattern in Sn(132)

Theorem

We let Qγ

n,132(q, x) = σ∈Sn(132) qcoinv(σ)xoccrγ(σ), then we have the

following recursive equations for the generating function Qγ

n,132(q, x).

Qγ

0,132(q, x)

= 1 for each pattern γ, (4) Q123

n,132(q, x)

=

n

• k=1

qk−1Qk−1(qx, x)Qn−k(q, x), (5) Q213

n,132(q, x)

=

n

• k=1

qk−1x

(k−1)(k−2) 2

Qk−1(q x , x)Qn−k(q, x), (6) Q231

n,132(q, x)

=

n

• k=1

qk−1x(k−1)(n−k)Qk−1(qx(n−k), x)Qn−k(q, x), (7) Q321

n,132(q, x)

=

n

• k=1

qk−1x

(n−k)(kn−4k+2) 2

Qk−1( q xn−k , x)Qn−k( q xk , x). (8)

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 19 / 28

Track all patterns of length 2 and 3 in Sn(132)

We can also track all the patterns that Q12,21,123,213,231,312,321

n,132

(x1, x2, x3, x4, x5, x6, x7) =

n

• k=1

xk−1

1

xk(n−k)

2

x(k−1)(n−k)

5

·Qk−1(x1x3x(n−k)

5

, x2x4x(n−k)

7

, x3, x4, x5, x6, x7) ·Qn−k(x1xk

6 , x2xk 7 , x3, x4, x5, x6, x7).

(9)

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 20 / 28

Track all patterns of length 2 and 3 in Sn(132)

Expansion of Q12,21,123,213,231,312,321

n,132

(x1, x2, x3, x4, x5, x6, x7)

n Q12,21,123,213,231,312,321

n,132

(x1, x2, x3, x4, x5, x6, x7) 1 1 1 2 x1 + x2 3 x3

1x7 + x2 1x2x5 + x2 1x2x6 + x1x2 2x4 + x3 2x3

4 x6

1x4 7 + x5 1x2x2 5x2 7 + x5 1x2x5x6x2 7 + x5 1x2x2 6x2 7 + x4 1x2 2x4x2 5x7 + x4 1x2 2x4x2 6x7 + x4 1x2 2x2 5x2 6

+x3

1x3 2x3x3 5 + x3 1x3 2x3x3 6 + x3 1x3 2x3 4x7 + x2 1x4 2x3x2 4x5 + x2 1x4 2x3x2 4x6 + x1x5 2x2 3x2 4 + x6 2x4 3

5 x10

1 x10 7 + x9 1x2x3 5x7 7 + x9 1x2x2 5x6x7 7 + x9 1x2x5x2 6x7 7 + x9 1x2x3 6x7 7 + x8 1x2 2x4x4 5x5 7 + x8 1x2 2x4x2 5x2 6x5 7

+x8

1x2 2x4x4 6x5 7 + x8 1x2 2x4 5x2 6x4 7 + x8 1x2 2x3 5x3 6x4 7 + x8 1x2 2x2 5x4 6x4 7 + x7 1x3 2x3x6 5x3 7 + x7 1x3 2x3x3 5x3 6x3 7

+x7

1x3 2x3x6 6x3 7 + x7 1x3 2x3 4x3 5x4 7 + x7 1x3 2x3 4x3 6x4 7 + x7 1x3 2x4x4 5x3 6x2 7 + x7 1x3 2x4x3 5x4 6x2 7 + x6 1x4 2x3x2 4x5 5x2 7

+x6

1x4 2x3x2 4x4 5x6x2 7 + x6 1x4 2x3x2 4x5x4 6x2 7 + x6 1x4 2x3x2 4x5 6x2 7 + x6 1x4 2x3x6 5x3 6 + x6 1x4 2x3x3 5x6 6

+x6

1x4 2x6 4x4 7 + x5 1x5 2x2 3x2 4x5 5x7 + x5 1x5 2x2 3x2 4x5 6x7 + x5 1x5 2x3x5 4x2 5x2 7 + x5 1x5 2x3x5 4x5x6x2 7

+x5

1x5 2x3x5 4x2 6x2 7 + x4 1x6 2x4 3x6 5 + x4 1x6 2x4 3x6 6 + x4 1x6 2x2 3x5 4x2 5x7 + x4 1x6 2x2 3x5 4x2 6x7 + x4 1x6 2x2 3x4 4x2 5x2 6

+x3

1x7 2x4 3x3 4x3 5 + x3 1x7 2x4 3x3 4x3 6 + x3 1x7 2x3 3x6 4x7 + x2 1x8 2x5 3x4 4x5 + x2 1x8 2x5 3x4 4x6 + x1x9 2x7 3x3 4 + x10 2 x10 3 Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 21 / 28

Counting Length 3 pattern in Sn(132)

We also get nice recursions for pattern distributions in Sn(123). For example, we have

Theorem

Let Q132

n,123(s, q, x) = σ∈Sn(123) sLRmin(σ)qcoinv(σ)xoccr132(σ), then we have

the following recursions, Q132

0,123(s, q, x)

= 1, Q132

n,123(s, q, x)

= sQn−1 +

n

• k=2

Qk−1(sq, qx, x)Qn−k(s, q, x).

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 22 / 28

An equality between Sn(132) and Sn(123)

We get nice recursions and functional equations for the function counting pattern 12 · · · m in Sn(132) and the function counting pattern 1m(m − 1) · · · 2 in Sn(123), for any m > 1. We found a big coincidence among Sn(132) and Sn(123) that, |{σ ∈ Sn(132) : occr12···j(σ) = i}| = |{σ ∈ Sn(123) : occr1j(j−1)···2(σ) = i}|, for all i < j.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 23 / 28

An equality between Sn(132) and Sn(123)

This result is described in the following theorem.

Theorem

We let Qn,132(x2, x3, . . . , xm) =

• σ∈Sn(132)

xoccr12

2

xoccr123

3

· · · xoccr12···m

m

, Q132(t, x2, x3, . . . , xm) =

• n≥0

tnQn,132(x2, x3, . . . , xm) and Qn,123(s, x2, x3, . . . , xm) =

• σ∈Sn(123)

sLRminxoccr12

2

xoccr132

3

· · · x

• ccr1m(m−1)···2

m

, Q123(t, s, x2, x3, . . . , xm) =

• n≥0

tnQn,123(s, x2, x3, . . . , xm),

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 24 / 28

Theorem

then we have the following equations, Qn,132(x2, . . . , xm) = n

k=1 xk−1 2

Qk−1,132(x2x3, x3x4, . . . , xm−1xm, xm)Qn−k,132(x2, . . . , xm), Qn,123(s, x2, . . . , xm) = sQn−1,123(t, s, x2, . . . , xm) + n

k=2 Qk−1,123(sx2, x2x3, x3x4, . . . , xm−1xm, xm)Qn−k,123(s, x2, . . . , xm),

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 25 / 28

Theorem

also the functional equations, Q132(t, x2, . . . , xm) = 1 + Q132(tx2, x2x3, x3x4, . . . , xm−1xm, xm)Q132(t, x2, . . . , xm), Q123(t, s, x2, . . . , xm) = 1 + t(s − 1)Q123(t, s, x2, . . . , xm) +tQ123(t, sx2, x2x3, x3x4, . . . , xm−1xm, xm)Q123(s, x2, . . . , xm). Further, let [xi]Q denote the coefficient of xi in function Q, then [tnxj

i ]Q132 = [tnxi j ]Q123 for i < j.

(10)

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 26 / 28

Outline

1

Motivation

2

Introduction

3

Wilf-equivalence of Qγ

λ(t, x) 4

Recursions of Qγ

λ(t, x) 5

Other Results and Open Problems

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 26 / 28

Other Results and Open Problems

We obtained the recursion tracking all patterns of length≤ 4 on Sn(132), see that every pattern is trackable on Sn(132).

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 27 / 28

Other Results and Open Problems

We obtained the recursion tracking all patterns of length≤ 4 on Sn(132), see that every pattern is trackable on Sn(132). On Sn(123), we only track patterns of length 2 and 3 and the special pattern 1m(m − 1) · · · 2. A simpler recursion on Sn(123) is desired.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 27 / 28

Other Results and Open Problems

We obtained the recursion tracking all patterns of length≤ 4 on Sn(132), see that every pattern is trackable on Sn(132). On Sn(123), we only track patterns of length 2 and 3 and the special pattern 1m(m − 1) · · · 2. A simpler recursion on Sn(123) is desired. We adapt our method to circular permutations. We track all circular patterns of size≤ 4 on circular permutations avoiding circular pattern 1243.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 27 / 28

Other Results and Open Problems

We obtained the recursion tracking all patterns of length≤ 4 on Sn(132), see that every pattern is trackable on Sn(132). On Sn(123), we only track patterns of length 2 and 3 and the special pattern 1m(m − 1) · · · 2. A simpler recursion on Sn(123) is desired. We adapt our method to circular permutations. We track all circular patterns of size≤ 4 on circular permutations avoiding circular pattern 1243. There are other equality of coefficients of generating functions Qγ

132

and Qγ

123 except equation (10) which we can study in the future.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 27 / 28

Other Results and Open Problems

We obtained the recursion tracking all patterns of length≤ 4 on Sn(132), see that every pattern is trackable on Sn(132). On Sn(123), we only track patterns of length 2 and 3 and the special pattern 1m(m − 1) · · · 2. A simpler recursion on Sn(123) is desired. We adapt our method to circular permutations. We track all circular patterns of size≤ 4 on circular permutations avoiding circular pattern 1243. There are other equality of coefficients of generating functions Qγ

132

and Qγ

123 except equation (10) which we can study in the future.

We only studied classical patterns on Sn(132) and Sn(123), and circular patterns on 1243.

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 27 / 28

Thank You!

Dun Qiu Pattern distribution in Sn(132) and Sn(123) July 11, 2018 28 / 28