´Enum´eration des intervalles du treillis de Tamari

(1)

Tamari lattice Counting intervals

Enum´ ´ eration des intervalles du treillis de Tamari

Viviane Pons

Universit¨ at Wien

LIAFA, 27 mars 2014

Viviane Pons ´Enum´eration des intervalles du treillis de Tamari

(2)

Definition on binary trees Intervals

Link with the weak order

Tamari lattice

I 2007, Chapoton : number of intervals 2

n(n + 1)

4n + 1 n − 1

(3)

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

(4)

Tamari lattice

I 1962, Tamari : poset of formal bracketing

I 1972, Huang, Tamari : lattice structure

n(n + 1) n − 1

(5)

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

(6)

m-Tamari lattices

number of intervals m + 1 n(mn + 1)

(m + 1) ² n + m n − 1

(7)

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

(8)

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

(9)

Binary trees

Recursive definition :

I the empty tree or

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

(10)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(11)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(12)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(13)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(14)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(15)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(16)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(17)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(18)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(19)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(20)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(21)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(22)

Right rotation

x y

A B

C →

x A y

B C

y x

→ x

y

(23)

(24)

Associahedron

(25)

1 2

3 4

5 6 7 8

9 10

Definition

An interval-poset is a poset of size n, labelled with 1, . . . , n such that

I if a < c and c C a then b C a for all a < b < c ,

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

We write a C b for a lower than b in the poset.

(26)

1 2

3 4

5 6 7 8

9 10

Definition

An interval-poset is a poset of size n, labelled with 1, . . . , n such that

I if a < c and c C a then b C a for all a < b < c ,

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

We write a C b for a lower than b in the poset.

(27)

1 2

3 4

5 6 7 8

9 10

Definition

An interval-poset is a poset of size n, labelled with 1, . . . , n such that

I if a < c and c C a then b C a for all a < b < c ,

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

We write a C b for a lower than b in the poset.

(28)

1 2

3 4

5 6 7 8

9 10

Definition

An interval-poset is a poset of size n, labelled with 1, . . . , n such that

I if a < c and c C a then b C a for all a < b < c ,

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

We write a C b for a lower than b in the poset.

(29)

1 2

3 4

5 6 7 8

9 10

Definition

An interval-poset is a poset of size n, labelled with 1, . . . , n such that

I if a < c and c C a then b C a for all a < b < c ,

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

We write a C b for a lower than b in the poset.

(30)

1 2

3 4

5 6 7 8

9 10

Theorem (Chˆ atel, P.)

Interval-posets are in bijections with intervals of the Tamari lattice.

(31)

Link with the weak order

(image from Jean-Louis Loday)

(32)

Binary search tree

x

< x > x

5 1

3 2 4

3 1

2 7

5 8

4 6

(33)

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, . . .

(34)

Binary search tree insertion

15324 →

4 2

1 3

5

(35)

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, . . .

(36)

Binary search tree insertion

15324 →

4 2

1

3

5

(37)

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, . . .

(38)

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, . . .

(39)

Binary search tree insertion

4 2

1 3

5 13254

31254 13524

31524 15324

35124 51324

53124

(40)

1234

2134 1324

3124

1243

1423

4123

2314 2143

2413

4213

1342

3142

3412

3214 2341 1432

4132

4312

3241 2431

4231 3421

4321

(41)

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

(42)

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 )

1 6

(43)

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

(44)

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 )

1 6

(45)

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

(46)

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

(47)

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

(48)

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

(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

2

1 3

4

(50)

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

(51)

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

(52)

1 2

4 3

5

10 8

7 9

6 1

5 2

4 3

7

6 10

8 9

(53)

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

(54)

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

(55)

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

(56)

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

(57)

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

(58)

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

(59)

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

(60)

Decomposition Smaller trees

1 3 2

5 6

4 7 8

x ⁴ = x ² .x. x ³ x ² z

3 x z

1 2 3

4 x ³ z ²

2

(61)

1 3 2

5 6

4 7 8

x ⁴ = x ² .x. x ³ x ² z

1 3 2

x ² z

1 2 3

4 x ³ z ²

2

(62)

1 3 2

5 6

4 7 8

x ⁴ = x ² .x. x ³ x ² z

1 3 2

1 2 3

4 x ³ z ²

2

(63)

1 3 2

5 6

4 7 8

x ⁴ = x ² .x. x ³ x ² z

1 3 2

x ² z

1 2 3

4 x ³ z ²

2

(64)

1 3 2

5 6

4 7 8

x ⁴ = x ² .x. x ³ x ² z

1 3 2

1 2 3

4 x ³ z ²

2

(65)

1 3 2

5 6

4 7 8 x ⁴

= x ² .x. x ³ x ² z

1 3 2

x ² z

1 2 3

4 x ³ z ²

2

(66)

1 3 2

5 6

4 7 8 x ⁴ = x ² .x. x ³

x ²

z

1 3 2

x ²

1 2 3

4 x ³

z ²

2

(67)

1 3 2

5 6

4 7 8 x ⁴ = x ² .x. x ³

x ²

z

1 3 2

x ²

z

1 2 3

4 x ³

z ²

2

(68)

1 3 2

5 6

4 7 8 x ⁴ = x ² .x. x ³

x ²

z

1 3 2

x ²

1 2 3

4 x ³

z ²

2

(69)

1 3 2

5 6

4 7 8 x ⁴ = x ² .x. x ³

x ² z

1 3 2

x ² z

1 2 3

4 x ³ z ²

2

(70)

0 1

1 3 2

5 6

4

7 8

1 3 2

5 6

4

7 8

x ⁶ x ⁵

2 3

1 3 2

5 6

4

7 8

1 3 2

5 6

4

7 8

x ⁴ x ³

(71)

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

(72)

1 3 2

= h

, i

x ²

1 2 3

4

=



 ,





x ³

5 6 7

8







,







(73)

1 3 2

= h

, i

x ²

1 2 3

4

=



 ,





x ³

1 3

2 4

5 6 7

8







,







(74)

1 3 2

= h

, i

x ²

1 2 3

4

=



 ,





x ³

1 3

2 4

5 6 7

8







,







(75)

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

(76)

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

(77)

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

(78)

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

(79)

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)

(80)

Tamari Polynomials B _T is recursively defined by

B _∅ := 1

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

avec 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.

(81)

Tamari Polynomials B _T is recursively defined by

B _∅ := 1

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

avec 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.

(82)

Tamari Polynomials B _T is recursively defined by

B _∅ := 1

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

avec 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.

(83)

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

(84)

B _∅ := 1

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

x − 1

B _L (x)= x ³ + x ²

(85)

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

(86)

B _∅ := 1

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

x − 1

B _R (x)= x ²

(87)

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

(88)

B _T (x) = x ³ + 2x ⁴ + 2x ⁵ + x ⁶

(89)

B _T (x) = x ³ + 2x ⁴ + 2x ⁵ + x ⁶

B _T (1) = 6

(90)

B _T (x) = x ³ + 2x ⁴ + 2x ⁵ + x ⁶

(91)

B _T (x) = x ³ + 2x ⁴ + 2x ⁵ + x ⁶

B _T (1) = 6

(92)

B _T (x) = x ³ + 2x ⁴ + 2x ⁵ + x ⁶

(93)

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

(94)

→

1 2

3

+

1 2

3

x ³ + x ²

→

¹ ²

x ²

x .x.x + x .x.x + x .x + x .x.x + x .x.x + x .x

(95)

→

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

(96)

→

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

x ³ .x.x ² + x ³ .x.x

+ x .x + x .x.x + x .x.x + x .x

(97)

→

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

(98)

→

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

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

+ x .x.x + x .x

(99)

→

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

(100)

→

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

(101)

→

1 2

3

+

1 2

3

x ³ + x ²

→

¹ ²

x ²

(x ³ + x ² ).x.(x ² + x + 1) =

(102)

→

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

´Enum´eration des intervalles du treillis de Tamari