Strobl,December16,2013 VivianePons Tamarilattice,rightweakorderandintervals

(1)

Some definitions Interval-posets Other results and perspectives

Tamari lattice, right weak order and intervals

Viviane Pons

Universit¨ at Wien

Strobl, December 16, 2013

Viviane Pons Tamari lattice, right weak order and intervals

(2)

Right weak order Tamari order

Definition

A permutation is a word of size n on the alphabet {1, . . . , n}

where each letter appears exactly once.

Example : 143592867

1 2 3 4 5 6 7 8 9

(3)

Definition

A permutation is a word of size n on the alphabet {1, . . . , n}

where each letter appears exactly once.

Example : 143592867

1 2 3 4 5 6 7 8 9

(4)

Definition

A permutation is a word of size n on the alphabet {1, . . . , n}

where each letter appears exactly once.

Example : 143592867

1 2 3 4 5 6 7 8 9

(5)

Group structure

σ = 143592867 µ = 324981756

1 2 3 4 5 6 7 8 9

µ.σ = 394862517

(6)

Group structure

σ = 143592867 µ = 324981756

1 2 3 4 5 6 7 8 9

µ.σ = 394862517

(7)

Group structure

σ = 143592867 µ = 324981756

1 2 3 4 5 6 7 8 9

µ.σ = 394862517

(8)

Transpositions

1 2 3 4 5

(2, 5)

Simple transpositions

1 2 3 4 5

(2, 3) = s 2

(9)

Transpositions

1 2 3 4 5

(2, 5) Simple transpositions

1 2 3 4 5

(2, 3) = s 2

(10)

Right weak order σ

σs _i

251436

521436 254136 251463

`(σs i ) = `(σ) + 1

(11)

Right weak order σ

σs _i

251436

521436 254136 251463

`(σs i ) = `(σ) + 1

(12)

Right weak order σ

σs _i

251436

521436 254136 251463

`(σs i ) = `(σ) + 1

(13)

Right weak order σ

σs _i

251436

521436 254136 251463

`(σs i ) = `(σ) + 1

(14)

Right weak order σ

σs _i

251436

521436 254136 251463

`(σs i ) = `(σ) + 1

(15)

Right weak order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

(16)

Right weak order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

(17)

Right weak order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

(18)

Right weak order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

(19)

Tamari lattice

I 1962, Tamari : poset of formal bracketing

I 1972, Huang, Tamari : lattice structure

I 2007, Chapoton : number of intervals 2

n(n + 1)

4n + 1 n − 1

(20)

Tamari lattice

I 1962, Tamari : poset of formal bracketing

I 1972, Huang, Tamari : lattice structure

I 2007, Chapoton : number of intervals 2

n(n + 1)

4n + 1 n − 1

(21)

Tamari lattice

I 1962, Tamari : poset of formal bracketing

I 1972, Huang, Tamari : lattice structure

I 2007, Chapoton : number of intervals 2

n(n + 1)

4n + 1 n − 1

(22)

Tamari lattice

I 1962, Tamari : poset of formal bracketing

I 1972, Huang, Tamari : lattice structure

I 2007, Chapoton : number of intervals 2

n(n + 1)

4n + 1 n − 1

(23)

m-Tamari lattices

I Bergeron, Pr´ eville-Ratelle : m-Tamari posets

I Bousquet-M´ elou, Fusy, Pr´ eville-Ratelle : lattice structure and number of intervals

m + 1 n(mn + 1)

(m + 1) ² n + m n − 1

(24)

m-Tamari lattices

I Bergeron, Pr´ eville-Ratelle : m-Tamari posets

I Bousquet-M´ elou, Fusy, Pr´ eville-Ratelle : lattice structure and number of intervals

m + 1 n(mn + 1)

(m + 1) ² n + m n − 1

(25)

m-Tamari lattices

I Bergeron, Pr´ eville-Ratelle : m-Tamari posets

I Bousquet-M´ elou, Fusy, Pr´ eville-Ratelle : lattice structure and number of intervals

m + 1 n(mn + 1)

(m + 1) ² n + m n − 1

(26)

Binary trees

Recursive definition :

I the empty tree or

I a left subtree and a right subtree grafted to a root node

Examples

(27)

Binary trees

Recursive definition :

I the empty tree or

I a left subtree and a right subtree grafted to a root node Examples

(28)

Right rotation

x y

A B

C →

x A y

B C

(29)

Right rotation

x y

A B

C →

x A y

B C

(30)

Right rotation

x y

A B

C →

x A y

B C

(31)

Right rotation

x y

A B

C →

x A y

B C

(32)

Right rotation

x y

A B

C →

x A y

B C

(33)

Right rotation

x y

A B

C →

x A y

B C

(34)

Right rotation

x y

A B

C →

x A y

B C

(35)

Right rotation

x y

A B

C →

x A y

B C

(36)

Right rotation

x y

A B

C →

x A y

B C

(37)

Right rotation

x y

A B

C →

x A y

B C

(38)

Right rotation

x y

A B

C →

x A y

B C

(39)

Right rotation

x y

A B

C →

x A y

B C

(40)

Right rotation

x y

A B

C →

x A y

B C

(41)

(42)

Link between the right weak order and the Tamari order canonical binary search tree labelling

x

< x > x

5 1

3 2 4

3 1

2 7

5 8

4 6

(43)

Binary search tree insertion

15324 →

4

2 1 3

5 Characterization : the permutations sent to a given tree are its linear extensions

15324, 31254, 35124, 51324, . . .

(44)

Binary search tree insertion

15324 →

4 2

1 3

5 Characterization : the permutations sent to a given tree are its linear extensions

15324, 31254, 35124, 51324, . . .

(45)

Binary search tree insertion

15324 →

4 2

1

3

5 Characterization : the permutations sent to a given tree are its linear extensions

15324, 31254, 35124, 51324, . . .

(46)

Binary search tree insertion

15324 →

4 2

1 3 5

Characterization : the permutations sent to a given tree are its linear extensions

15324, 31254, 35124, 51324, . . .

(47)

Binary search tree insertion

15324 →

4 2

1 3

5 Characterization : the permutations sent to a given tree are its linear extensions

15324, 31254, 35124, 51324, . . .

(48)

Binary search tree insertion

15324 →

4 2

1 3

5 Characterization : the permutations sent to a given tree are its linear extensions

15324, 31254, 35124, 51324, . . .

(49)

1234

2134 1324

3124

1243

1423

4123

2314 2143

2413

4213

1342

3142

3412

3214 2341 1432

4132

4312

3241 2431

4231 3421

4321

(50)

Construction Counting intervals Composition of interval-posets

4 2 5

1 3 9

7 6 8

final forest F ≥ (T )

1 2

3 4 5

6 7 9

8 Initial forest F _≤ (T )

4 2 3

1 5 9

7 8

6

(51)

4 2 5

1 3 9

7 6 8

final forest F ≥ (T )

1 2

3 4 5

6 7 9

8 Initial forest F _≤ (T )

4 2 3

1 5 9

7 8

6

(52)

4 2 5

1 3 9

7 6 8

final forest F ≥ (T )

1 2

3 4 5

6 7 9

8 Initial forest F _≤ (T )

4 2 3

1 5 9

7 8

6

(53)

4 2 5

1 3 9

7 6 8

final forest F ≥ (T )

1 2

3 4 5

6 7 9

8 Initial forest F _≤ (T )

4 2 3

1 5 9

7 8

6

(54)

4 2 5

1 3 9

7 6 8

final forest F ≥ (T )

1 2

3 4 5

6 7 9

8 Initial forest F _≤ (T )

4 2 3

1 5 9

7 8

6

(55)

1234

2134 1324

3124

1243

1423

4123

2314 2143

2413

4213

1342

3142

3412

3214 2341 1432

4132

4312

3241 2431

4231 3421

4321

4 2

1 3

(56)

1234

2134 1324

3124

1243

1423

4123

2314 2143

2413

4213

1342

3142

3412

3214 2341 1432

4132

4312

3241 2431

4231 3421

4321

F ≤ (T )

1

2 3

4

(57)

1234

2134 1324

3124

1243

1423

4123

2314 2143

2413

4213

1342

3142

3412

3214 2341 1432

4132

4312

3241 2431

4231 3421

4321

F ≥ (T )

1 2

3 4

(58)

1234

2134 1324

3124

1243

1423

4123

2314 2143

2413

4213

1342

3142

3412

3214 2341 1432

4132

4312

3241 2431

4231 3421

4321

2

1 3

4

(59)

1234

2134 1324

3124

1243

1423

4123

2314 2143

2413

4213

1342

3142

3412

3214 2341 1432

4132

4312

3241 2431

4231 3421

4321

F ≤ (T ⁰ )

1

2 3 4

(60)

1234

2134 1324

3124

1243

1423

4123

2314 2143

2413

4213

1342

3142

3412

3214 2341 1432

4132

4312

3241 2431

4231 3421

4321

F ≥ (T )

1 2

3 4

F ≤ (T ⁰ )

1

2 3 4

Intervalle-poset [T , T ⁰ ]

1 2

3 4

(61)

1 2

4 3

5

10 8

7 9

6 1

5 2

4 3

7

6 10

8 9

(62)

1 2

4 3

5

10 8

7 9

6 1

5 2

4 3

7

6 10

8 9

1 2

3 4

5 6 7 8

9 10

1 2

3 4 5

6 7

8 9

10

(63)

1 2

4 3

5

10 8

7 9

6 1

5 2

4 3

7

6 10

8 9

1 2

3 4

5 6 7 8

9 10

(64)

1 2

4 3

5

10 8

7 9

6 1

5 2

4 3

7

6 10

8 9

1 2

3 4

5 6 7 8

9 10

(65)

1 2

4 3

5

10 8

7 9

6 1

5 2

4 3

7

6 10

8 9

1 2

3 4

5 6 7 8

9 10

(66)

1 2

4 3

5

10 8

7 9

6 1

5 2

4 3

7

6 10

8 9

1 2

3 4

5 6 7 8

9 10

(67)

1 2

4 3

5

10 8

7 9

6 1

5 2

4 3

7

6 10

8 9

1 2

3 4

5 6 7 8

9 10

(68)

1 2

4 3

5

10 8

7 9

6

1

5

2 4 3

7

6 10

8 9

1 2

3 4

5 6 7 8

9 10

(69)

Theorem (Chˆ atel, P.)

Intervals of Tamari are in bijection with labelled posets of size n and labels 1, . . . , n such that

I If a < c and a C c then b C c for all a < b < c.

I If a < c and c C a then b C a for all a < b < c.

1 4

⇒

1 2 3

4 4 1

⇒

4 3 2

1

(70)

Theorem (Chˆ atel, P.)

Intervals of Tamari are in bijection with labelled posets of size n and labels 1, . . . , n such that

I If a < c and a C c then b C c for all a < b < c.

I If a < c and c C a then b C a for all a < b < c.

1 4

⇒

1 2 3

4 4 1

⇒

4 3 2

1

(71)

Theorem (Chˆ atel, P.)

Intervals of Tamari are in bijection with labelled posets of size n and labels 1, . . . , n such that

I If a < c and a C c then b C c for all a < b < c.

I If a < c and c C a then b C a for all a < b < c.

1 4

⇒

1 2 3

4 4 1

⇒

4 3 2

1

(72)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2 + 2 + 1 = 13

(2x + 2x ² + x ³ )

+ (x + x ² + x ³ ) + (x ² + x ³ ) + (x ² + x ³ ) + x ³

(73)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2 + 2 + 1 = 13

(2x + 2x ² + x ³ )

+ (x + x ² + x ³ ) + (x ² + x ³ ) + (x ² + x ³ ) + x ³

(74)

x

³

x

²

x

²

x

Number of intervals

5 + 3

+ 2 + 2 + 1 = 13

(2x + 2x ² + x ³ )

+ (x + x ² + x ³ ) + (x ² + x ³ ) + (x ² + x ³ ) + x ³

(75)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2

+ 2 + 1 = 13

(2x + 2x ² + x ³ )

+ (x + x ² + x ³ ) + (x ² + x ³ ) + (x ² + x ³ ) + x ³

(76)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2 + 2

+ 1 = 13

(2x + 2x ² + x ³ )

+ (x + x ² + x ³ ) + (x ² + x ³ ) + (x ² + x ³ ) + x ³

(77)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2 + 2 + 1

= 13

(2x + 2x ² + x ³ )

+ (x + x ² + x ³ ) + (x ² + x ³ ) + (x ² + x ³ ) + x ³

(78)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2 + 2 + 1 = 13

(2x + 2x ² + x ³ )

+ (x + x ² + x ³ ) + (x ² + x ³ ) + (x ² + x ³ ) + x ³

(79)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2 + 2 + 1 = 13

(2x + 2x ² + x ³ )

+ (x + x ² + x ³ ) + (x ² + x ³ ) + (x ² + x ³ ) + x ³

(80)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2 + 2 + 1 = 13

(2x + 2x ² + x ³ )

+ (x + x ² + x ³ ) + (x ² + x ³ ) + (x ² + x ³ ) + x ³

(81)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2 + 2 + 1 = 13

(2x + 2x ² + x ³ ) + (x + x ² + x ³ )

+ (x ² + x ³ ) + (x ² + x ³ ) + x ³

(82)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2 + 2 + 1 = 13

(2x + 2x ² + x ³ ) + (x + x ² + x ³ ) + (x ² + x ³ )

+ (x ² + x ³ ) + x ³

(83)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2 + 2 + 1 = 13

(2x + 2x ² + x ³ ) + (x + x ² + x ³ ) + (x ² + x ³ ) + (x ² + x ³ )

+ x ³

(84)

x

³

x

²

x

²

x

Number of intervals

5 + 3 + 2 + 2 + 1 = 13

(2x + 2x ² + x ³ ) + (x + x ² + x ³ ) + (x ² + x ³ ) + (x ² + x ³ ) + x ³

(85)

Theorem (Chapoton)

The generating functions of Tamari intervals satisfy the functional equation

Φ(x, y ) = B(Φ, Φ) + 1 where

B(f , g ) = xyf (x, y) xg (x, y ) − g (1, y) x − 1

(86)

Tamari Polynomials B _T is recursively defined by

B _∅ := 1

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

with T =

L

• R

Theorem (Chˆ atel, P.)

B _T counts the number of trees smaller than or equal to T in the Tamari lattice according to the number of nodes on their left border.

(87)

Tamari Polynomials B _T is recursively defined by

B _∅ := 1

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

with T =

L

• R

Theorem (Chˆ atel, P.)

B _T counts the number of trees smaller than or equal to T in the Tamari lattice according to the number of nodes on their left border.

(88)

Tamari Polynomials B _T is recursively defined by

B _∅ := 1

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

with T =

L

• R

Theorem (Chˆ atel, P.)

B _T counts the number of trees smaller than or equal to T in the Tamari lattice according to the number of nodes on their left border.

(89)

B _∅ := 1

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

B _L (x)= x ³ + x ² B _R (x)= x ²

(90)

B _∅ := 1

B _T (x) := xB _L (x) xB _R (x) − B _R (1) x − 1 B _L (x)= x ³ + x ²

B _R (x)= x ²

(91)

B _∅ := 1

B _T (x) := xB _L (x) xB _R (x) − B _R (1) x − 1 B _L (x)= x ³ + x ²

B _R (x)= x ²

(92)

B _∅ := 1

B _T (x) := x(x ³ + x ² ) xB _R (x) − B _R (1) x − 1

B _L (x)= x ³ + x ²

B _R (x)= x ²

(93)

B _∅ := 1

B _T (x ) := x(x ³ + x ² )(1 + x + x ² )

B _L (x)= x ³ + x ² B _R (x)= x ²

(94)

B _T (x) = x ³ + 2x ⁴ + 2x ⁵ + x ⁶

B _T (1) = 6

(95)

B _T (x) = x ³ + 2x ⁴ + 2x ⁵ + x ⁶

B _T (1) = 6

(96)

B _T (x) = x ³ + 2x ⁴ + 2x ⁵ + x ⁶

B _T (1) = 6

(97)

B _T (x) = x ³ + 2x ⁴ + 2x ⁵ + x ⁶

B _T (1) = 6

(98)

B _T (x) = x ³ + 2x ⁴ + 2x ⁵ + x ⁶

B _T (1) = 6

(99)

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

(100)

B(

1 3 2

, 1 2 3

4 ) =

1 3 2

4

5 6 7

8

+ 1

3

2 4

5 6 7

8

+ 1

3

2 4

5 6 7

8 +

1 3

2 4

5 6 7

8

(101)

B(

1 3 2

, 1 2 3

4 ) =

1 3 2

4

5 6 7

8 +

1 3

2 4

5 6 7

8

+ 1

3

2 4

5 6 7

8 +

1 3

2 4

5 6 7

8

(102)

B(

1 3 2

, 1 2 3

4 ) =

1 3

2 4

5 6 7

8 +

1 3

2 4

5 6 7

8

+ 1

3

2 4

5 6 7

8 +

1 3

2 4

5 6 7

8

(103)

B(

1 3 2

, 1 2 3

4 ) =

1 3

2 4

5 6 7

8

+ 1

3

2 4

5 6 7

8

+ 1

3

2 4

5 6 7

8 +

1 3

2 4

5 6 7

8

(104)

B(

1 3 2

, 1 2 3

4 ) =

1 3

2 4

5 6 7

8 +

1 3

2 4

5 6 7

8

+ 1

3

2 4

5 6 7

8 +

1 3

2 4

5 6 7

8

(105)

B(

1 3 2

, 1 2 3

4 ) =

1 3

2 4

5 6 7

8 +

1 3

2 4

5 6 7

8

+ 1

3

2 4

5 6 7

8

+ 1

3

2 4

5 6 7

8

(106)

B(

1 3 2

, 1 2 3

4 ) =

1 3

2 4

5 6 7

8 +

1 3

2 4

5 6 7

8

+ 1

3

2 4

5 6 7

8 +

1 3

2 4

5 6 7

8

(107)

1 3 2

= h

, i

x ²

1 2 3

4

=



 ,





x ³

1 3

2 4

5 6 7

8







,







(108)

1 3 2

= h

, i

x ²

1 2 3

4

=



 ,





x ³

1 3

2 4

5 6 7

8







,







(109)

1 3 2

= h

, i

x ²

1 2 3

4

=



 ,





x ³

1 3

2 4

5 6 7

8







,







(110)

1 3 2

= h

, i

x ²

1 2 3

4

=



 ,





x ³

1 3

2 4

5 6 7

8







,







x ² .x.x ³

(111)

1 3 2

= h

, i

x ²

1 2 3

4

=



 ,





x ³

1 3

2 4

5 6 7

8







,







x ² .x.x ³ + x ² .x.x ²

(112)

1 3 2

= h

, i

x ²

1 2 3

4

=



 ,





x ³

1 3

2 4

5 6 7

8





 ,







x ² .x.x ³ + x ² .x.x ² + x ² .x.x

(113)

1 3 2

= h

, i

x ²

1 2 3

4

=



 ,





x ³

1 3

2 4

5 6 7

8





 ,







x ² .x.x ³ + x ² .x.x ² + x ² .x.x + x ² .x

(114)

1 3 2

= h

, i

x ²

1 2 3

4

=



 ,





x ³

1 3

2 4

5 6 7

8





 ,







x ² .x.(x ³ + x ² + x + 1)

(115)

S _T := X

T

⁰

≤T

[T ⁰ , T ] S _T = B (S _L , S _R )

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

(116)

→

1 2

3

+

1 2

3

x ³ + x ²

→

¹ ²

x ²

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

x ³ .x.x ² + x ³ .x.x + x ³ .x + x ² .x.x ² + x ² .x.x + x ² .x

(117)

→

1 2

3

+

1 2

3

x ³

+ x ²

→

¹ ²

x ²

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

x ³ .x.x ²

+ x ³ .x.x + x ³ .x + x ² .x.x ² + x ² .x.x + x ² .x

(118)

→

1 2

3

+

1 2

3

x ³

+ x ²

→

¹ ²

x ²

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

x ³ .x.x ² + x ³ .x.x

+ x ³ .x + x ² .x.x ² + x ² .x.x + x ² .x

(119)

→

1 2

3

+

1 2

3

x ³

+ x ²

→

¹ ²

x ²

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

x ³ .x.x ² + x ³ .x.x + x ³ .x

+ x ² .x.x ² + x ² .x.x + x ² .x

(120)

→

1 2

3

+

1 2

3

x ³ + x ²

→

¹ ²

x ²

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

x ³ .x.x ² + x ³ .x.x + x ³ .x + x ² .x.x ²

+ x ² .x.x + x ² .x

(121)

→

1 2

3

+

1 2

3

x ³ + x ²

→

¹ ²

x ²

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

x ³ .x.x ² + x ³ .x.x + x ³ .x + x ² .x.x ² + x ² .x.x

+ x ² .x

(122)

→

1 2

3

+

1 2

3

x ³ + x ²

→

¹ ²

x ²

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

+

1 2

3 4

5 6

x ³ .x.x ² + x ³ .x.x + x ³ .x + x ² .x.x ² + x ² .x.x + x ² .x

(123)

→

1 2

3

+

1 2

3

x ³ + x ²

→

¹ ²

x ²

(x ³ + x ² ).x.(x ² + x + 1) =

(124)

→

1 2

3

+

1 2

3

x ³ + x ²

→

¹ ²

x ²

(x ³ + x ² ).x.(x ² + x + 1) = x B _L (x) B _R (x) − B _R (1) x − 1

(125)