2 Our work

Introduction Our work

Counting smaller trees in the Tamari order

Gr´ egory Chatel, Viviane Pons

April 27, 2015

Introduction Our work

1 Introduction Basic definitions Tamari lattice Goal

2 Our work

Main result

Example

Sketch of proof

Introduction Our work

Basic definitions Tamari lattice Goal

Permutation and weak order

Permutations

A permutation σ is a word of size n where every letter of {1, . . . , n} appear only once.

Ex : 1234, 15234, 4231.

Weak order : partial order on permutations

At each step, we inverse two increasing consecutives values.

123 213 132

231

312

Introduction Our work

Basic definitions Tamari lattice Goal

Binary trees

Binary trees

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Canonical labelling

x

5 1

3

1 7

Introduction Our work

Basic definitions Tamari lattice Goal

Permutation to binary tree

Binary search tree insertion

15324 →

4

2

1 3

5

Introduction Our work

Basic definitions Tamari lattice Goal

Permutation to binary tree

Binary search tree insertion

15324 →

4 2

1 3

5

Introduction Our work

Basic definitions Tamari lattice Goal

Permutation to binary tree

Binary search tree insertion

15324 →

4 2

1

3

5

Introduction Our work

Basic definitions Tamari lattice Goal

Permutation to binary tree

Binary search tree insertion

15324 →

4 2

1

3

5

Introduction Our work

Basic definitions Tamari lattice Goal

Permutation to binary tree

Binary search tree insertion

15324 →

4 2

1 3

5

Introduction Our work

Basic definitions Tamari lattice Goal

Order relation

Right rotation

x y

A B

C →

x A y

B C

This gives a partial order on binary trees.

Example

4 4

4

Introduction Our work

Basic definitions Tamari lattice Goal

Tamari lattice

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Figure : Tamari lattices of size 3 and 4.

Introduction Our work

Basic definitions Tamari lattice Goal

Tamari lattice as a quotient of the weak order

123 213 132

231 312

1 2

3

1 2

3 1

2 3

1

2 3

1 2

2 4

5

13254

31254 13524

31524 15324

35124 51324

Introduction Our work

Basic definitions Tamari lattice Goal

What do we want to do ?

Goal

We want a formula that computes, for any given tree T the number of trees smaller than T in the Tamari order.

Example : how many trees are smaller than this one ?

1 2 3

4

5

6

Introduction Our work

Main result Example Sketch of proof

1 Introduction Basic definitions Tamari lattice Goal

2 Our work

Main result

Example

Sketch of proof

Introduction Our work

Main result Example Sketch of proof

Tamari polynomials

Tamari polynomials

Given a binary tree T , we define:

B ∅ := 1 (1)

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1)

x − 1 (2)

with T =

L

•

R

Introduction Our work

Main result Example Sketch of proof

Main theorem

Theorem

Let T be a binary tree. Its Tamari polynomial B _T (x) counts

the trees smaller than T in the Tamari order according to the

number of nodes on their left border. In particular, B _T (1) is

the number of trees smaller than T .

Introduction Our work

Main result Example Sketch of proof

Example

B ∅ := 1

B T (x ) := x B L (x ) x B R (x ) − B R (1) x − 1

1 2 3

4

5

6

Introduction Our work

Main result Example Sketch of proof

Example

B _∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5 6

• B _T (x ) = B ₄ (x) = xB ₃ (x ) ^B

^(x)−B _x−1

⁽¹⁾

• B ₃ (x ) = x B ₁ (x )

• B ₁ (x ) = x ^B

^(x)−B _x−1

⁽¹⁾

• B ₂ (x ) = x

• B 1 (x ) = x (1 + x ) = x + x ²

• B ₃ (x ) = x (x + x ² ) = x ² + x ³

Introduction Our work

Main result Example Sketch of proof

Example

B _∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5 6

• B _T (x ) = B ₄ (x) = x B ₃ (x ) ^B

^(x)−B _x−1

⁽¹⁾

• B ₃ (x ) = x B ₁ (x )

• B ₁ (x ) = x ^B

^(x)−B _x−1

⁽¹⁾

• B ₂ (x ) = x

• B 1 (x ) = x (1 + x ) = x + x ²

• B ₃ (x ) = x (x + x ² ) = x ² + x ³

Introduction Our work

Main result Example Sketch of proof

Example

B _∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5 6

• B _T (x ) = B ₄ (x) = x B ₃ (x ) ^B

^(x)−B _x−1

⁽¹⁾

• B ₃ (x ) = x B ₁ (x )

• B ₁ (x ) = x ^B

^(x)−B _x−1

⁽¹⁾

• B ₂ (x ) = x

• B 1 (x ) = x (1 + x ) = x + x ²

• B ₃ (x ) = x (x + x ² ) = x ² + x ³

Introduction Our work

Main result Example Sketch of proof

Example

B _∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5 6

• B _T (x ) = B ₄ (x) = x B ₃ (x ) ^B

^(x)−B _x−1

⁽¹⁾

• B ₃ (x ) = x B ₁ (x )

• B ₁ (x ) = x ^B

^(x)−B _x−1

⁽¹⁾

• B 2 (x ) = x

• B 1 (x ) = x (1 + x ) = x + x ²

• B ₃ (x ) = x (x + x ² ) = x ² + x ³

Introduction Our work

Main result Example Sketch of proof

Example

B _∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5 6

• B _T (x ) = B ₄ (x) = x B ₃ (x ) ^B

^(x)−B _x−1

⁽¹⁾

• B ₃ (x ) = x B ₁ (x )

• B ₁ (x ) = x ^B

^(x)−B _x−1

⁽¹⁾

• B 2 (x ) = x

• B ₁ (x ) = x (1 + x ) = x + x ²

• B ₃ (x ) = x (x + x ² ) = x ² + x ³

Introduction Our work

Main result Example Sketch of proof

Example

B _∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5 6

• B _T (x ) = B ₄ (x) = x B ₃ (x ) ^B

^(x)−B _x−1

⁽¹⁾

• B ₃ (x ) = x B ₁ (x )

• B ₁ (x ) = x ^B

^(x)−B _x−1

⁽¹⁾

• B 2 (x ) = x

• B ₁ (x ) = x (1 + x ) = x + x ²

• B ₃ (x ) = x (x + x ² ) = x ² + x ³

Introduction Our work

Main result Example Sketch of proof

Example

B ∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5

6 • B T (x ) = B 4 (x) = x(x ² + x ³ ) ^B

^(x)−B _x−1

⁽¹⁾

• B 6 (x ) = x B 5 (x )

• B ₅ (x ) = x

• B ₆ (x ) = xx = x ²

Introduction Our work

Main result Example Sketch of proof

Example

B ∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5

6 • B T (x ) = B 4 (x) = x(x ² + x ³ ) ^B

^(x)−B _x−1

⁽¹⁾

• B ₆ (x ) = x B ₅ (x )

• B ₅ (x ) = x

• B ₆ (x ) = xx = x ²

Introduction Our work

Main result Example Sketch of proof

Example

B ∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5

6 • B T (x ) = B 4 (x) = x(x ² + x ³ ) ^B

^(x)−B _x−1

⁽¹⁾

• B ₆ (x ) = x B ₅ (x )

• B ₅ (x ) = x

• B ₆ (x ) = xx = x ²

Introduction Our work

Main result Example Sketch of proof

Example

B ∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5

6 • B T (x ) = B 4 (x) = x(x ² + x ³ ) ^B

^(x)−B _x−1

⁽¹⁾

• B ₆ (x ) = x B ₅ (x )

• B ₅ (x ) = x

• B ₆ (x ) = xx = x ²

Introduction Our work

Main result Example Sketch of proof

Example

B _∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5 6

• B ₄ (x ) = x (x ² + x ³ )(1 + x + x ² )

• B ₄ (x ) = x ⁶ + 2x ⁵ + 2x ⁴ + x ³

Introduction Our work

Main result Example Sketch of proof

Example

B _∅ := 1

B _T (x ) := x B _L (x ) x B _R (x ) − B _R (1) x − 1

1 2 3

4

5 6

• B ₄ (x ) = x (x ² + x ³ )(1 + x + x ² )

• B ₄ (x ) = x ⁶ + 2x ⁵ + 2x ⁴ + x ³

Introduction Our work

Main result Example Sketch of proof

Example

1 2 3

4 5

6 B _T (x) = x ³ + 2x ⁴ + 2x ⁵ + x ⁶ B _T (1) = 6

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Introduction Our work

Main result Example Sketch of proof

Sketch of proof

Increasing and decreasing forests

4

2 5

1 3 9

7

6 8

Introduction Our work

Main result Example Sketch of proof

Sketch of proof

Increasing and decreasing forests

4

2 5

1 3 9

7

6 8

Introduction Our work

Main result Example Sketch of proof

Sketch of proof

Increasing and decreasing forests

4

2 5

1 3 9

7

6 8

1 2

3

4 5

6 7 9

8

Introduction Our work

Main result Example Sketch of proof

Sketch of proof

Increasing and decreasing forests

4

2 5

1 3 9

7

6 8

4

2 3

1

5 9

7 8

6

Introduction Our work

Main result Example Sketch of proof

Initial and final intervals

1 2

3

1 2

3 1

2 3

1

2 3

1 2

3

1

2 3

123 213 132

231 312

321

Introduction Our work

Main result Example Sketch of proof

Initial and final intervals

1 2

3

1 2

3 1

2 3

1

2 3

1 2

3

1 2 3

123 213 132

231 312

321

Introduction Our work

Main result Example Sketch of proof

Initial and final intervals

1 2

3

1 2

3 1

2 3

1

2 3

1 2

3

1

2 3

123 213 132

231 312

321

Introduction Our work

Main result Example Sketch of proof

Linear extensions of any interval

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Introduction Our work

Main result Example Sketch of proof

Linear extensions of any interval

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2 3

4

Introduction Our work

Main result Example Sketch of proof

Linear extensions of any interval

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1 2

3

4

Introduction Our work

Main result Example Sketch of proof

Linear extensions of any interval

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2 3

4

Introduction Our work

Main result Example Sketch of proof

Sketch of proof

Bilinear form

B T (x ) = xB L (x) x B _R (x ) − B _R (1) x − 1

B(f , g ) = xf (x ) xg(x ) − g(1)

x − 1

Introduction Our work

Main result Example Sketch of proof

Sketch of proof

Combinatorial interpretation

X

T

<T

[T ⁰ , T ] = B ( X

T1

<T1

[T 1 ⁰ , T 1], X

T2

<T2

[T 2 ⁰ , T 2])

with T =

T 1

•

T 2

Introduction Our work

Main result Example Sketch of proof

B (

1 2

3

, 2

1 3

) =

1 2

3

4

1 2

3

+ 1 2

3 4

5 6

7

+ 1 2

3 4

5 6

7

Introduction Our work

Main result Example Sketch of proof

B (

1 2

3

, 2

1 3

) =

1 2

3 4

5 6

7

+ 1 2

3 4

5 6

7

+ 1 2

3 4

5 6

7

Introduction Our work

Main result Example Sketch of proof

B (

1 2

3

, 2

1 3

) =

1 2

3 4

5 6

7

+ 1 2

3 4

5 6

7

+ 1 2

3 4

5 6

7

Introduction Our work

Main result Example Sketch of proof

B (

1 2

3

, 2

1 3

) =

1 2

3 4

5 6

7

+ 1 2

3 4

5 6

7

+ 1 2

3 4

5 6

7

Introduction Our work

Main result Example Sketch of proof

B (

1 2

3

, 2

1 3

) =

1 2

3 4

5 6

7

+ 1 2

3 4

5 6

7

+ 1 2

3 4

5 6

7

Introduction Our work

Main result Example Sketch of proof

1 2 3

4 5

6

B ( 3

1 2

+

3 1

2

, 2

1 )

[

3 2 1

, 3

2

1 ] [

3 2

1 ,

3 2

1 ] [ 2

1

, 2

1 ]

Introduction Our work

Main result Example Sketch of proof

B ( 3

1 2 +

3 1

2

, 2

1 ) = 4

3 1 2

6 5

+

4 3 1 2

6 5

+

4 3 1 2

6 5

+

4 4 4

Introduction Our work

Main result Example Sketch of proof

4 3 1 2

6 5

+

4 3 1 2

6 5

+

4 3 1 2

6 5

+

4 3 1 2

6

5 +

4 3 1 2

6

5 +

4 3 1 2

6

5

Introduction Our work

Main result Example Sketch of proof

[

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x ⁶

+

+ [

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x ⁵

+

+

[ _• ^•

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x ⁴

+

+

[

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Introduction Our work

Main result Example Sketch of proof

Example

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Introduction Our work

Main result Example Sketch of proof

Questions ?