Tamari lattice Counting intervals
Intervals of the Tamari lattice
Viviane Pons
Universit¨ at Wien
UQAM, May 2nd, 2014
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
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
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Tamari lattice
I 1962, Tamari : poset of formal bracketing
I 1972, Huang, Tamari : lattice structure
n(n + 1) n − 1
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
m-Tamari lattices
number of intervals m + 1 n(mn + 1)
(m + 1) 2 n + m n − 1
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
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
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Binary trees
Recursive definition :
I the empty tree or
I a left subtree and a right subtree grafted to a root node Examples
Viviane Pons Intervals of the Tamari lattice
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Viviane Pons Intervals of the Tamari lattice
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Viviane Pons Intervals of the Tamari lattice
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Viviane Pons Intervals of the Tamari lattice
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Viviane Pons Intervals of the Tamari lattice
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Viviane Pons Intervals of the Tamari lattice
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Viviane Pons Intervals of the Tamari lattice
Right rotation
x y
A B
C →
x A y
B C
y x
→ x
y
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Viviane Pons Intervals of the Tamari lattice
Associahedron
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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.
Viviane Pons Intervals of the Tamari lattice
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.
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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.
Viviane Pons Intervals of the Tamari lattice
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.
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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.
Viviane Pons Intervals of the Tamari lattice
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.
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Link with the weak order
(image from Jean-Louis Loday)
Viviane Pons Intervals of the Tamari lattice
Binary search tree
x
< x > x
5 1
3
2 4
3 1
2 7
5 8
4 6
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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, . . .
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Binary search tree insertion
15324 →
4 2
1 3
5
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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, . . .
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Binary search tree insertion
15324 →
4 2
1
3
5
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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, . . .
Viviane Pons Intervals of the Tamari lattice
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, . . .
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
Binary search tree insertion
4 2
1 3
5
13254
31254 13524
31524 15324
35124 51324
53124
Viviane Pons Intervals of the Tamari lattice
1234
2134 1324
3124
1243
1423
4123
2314 2143
2413
4213
1342
3142
3412
3214 2341 1432
4132
4312
3241 2431
4231 3421
4321
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
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
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
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
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
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
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
1 2
4 3
5
10 8
7 9
6 1
5 2
4 3
7
6 10
8 9
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
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
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
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
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
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
Tamari lattice Counting intervals
Definition on binary trees Intervals
Link with the weak order
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
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Decomposition Smaller trees
1 3 2
5 6
4
7 8
x 4 = x 2 .x. x 3 x 2
3 x
1 2 3
4 x 3
2
Tamari lattice Counting intervals
Decomposition Smaller trees
1 3 2
5 6
4
7 8
x 4 = x 2 .x. x 3 x 2
1 3 2
x 2
1 2 3
4 x 3
2
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Decomposition Smaller trees
1 3 2
5 6
4
7 8
x 4 = x 2 .x. x 3 x 2
1 3 2
1 2 3
4 x 3
2
Tamari lattice Counting intervals
Decomposition Smaller trees
1 3 2
5 6
4
7 8
x 4 = x 2 .x. x 3 x 2
1 3 2
x 2
1 2 3
4
x 3
2
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Decomposition Smaller trees
1 3 2
5 6
4
7 8
x 4 = x 2 .x. x 3 x 2
1 3 2
1 2 3
4
x 3
2
Tamari lattice Counting intervals
Decomposition Smaller trees
1 3 2
5 6
4
7 8 x 4
= x 2 .x. x 3 x 2
1 3 2
x 2
1 2 3
4
x 3
2
Viviane Pons Intervals of the Tamari lattice
1 3 2
5 6
4
7 8 x 4 = x 2 .x. x 3
x 2
1 3 2
x 2
1 2 3
4 x 3
2
Tamari lattice Counting intervals
Decomposition Smaller trees
1 3 2
5 6
4
7 8 x 4 = x 2 .x. x 3
x 2
1 3 2
x 2
1 2 3
4 x 3
2
Viviane Pons Intervals of the Tamari lattice
1 3 2
5 6
4
7 8 x 4 = x 2 .x. x 3
x 2
1 3 2
x 2
1 2 3
4 x 3
2
Tamari lattice Counting intervals
Decomposition Smaller trees
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
Viviane Pons Intervals of the Tamari lattice
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
Tamari lattice Counting intervals
Decomposition Smaller trees
1 3 2
= h
, i
x 2
1 2 3
4
=
,
x 3
1 3
2 4
5 6 7
8
,
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Decomposition Smaller trees
1 3 2
= h
, i
x 2
1 2 3
4
=
,
x 3
5 6 7
8
,
Tamari lattice Counting intervals
Decomposition Smaller trees
1 3 2
= h
, i
x 2
1 2 3
4
=
,
x 3
1 3
2 4
5 6 7
8
,
Viviane Pons Intervals of the Tamari lattice
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
Tamari lattice Counting intervals
Decomposition Smaller trees
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
Viviane Pons Intervals of the Tamari lattice
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
Tamari lattice Counting intervals
Decomposition Smaller trees
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
Viviane Pons Intervals of the Tamari lattice
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)
Tamari lattice Counting intervals
Decomposition Smaller trees
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.
Viviane Pons Intervals of the Tamari lattice
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.
Tamari lattice Counting intervals
Decomposition Smaller trees
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.
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Decomposition Smaller trees
B ∅ := 1
B T (x) := xB L (x) xB R (x) − B R (1)
x − 1
Tamari lattice Counting intervals
Decomposition Smaller trees
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
Viviane Pons Intervals of the Tamari lattice
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
Tamari lattice Counting intervals
Decomposition Smaller trees
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
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Decomposition Smaller trees
B ∅ := 1
B T (x ) := x(x 3 + x 2 )(1 + x + x 2 )
Tamari lattice Counting intervals
Decomposition Smaller trees
B T (x) = x 3 + 2x 4 + 2x 5 + x 6
B T (1) = 6
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Decomposition Smaller trees
B T (x) = x 3 + 2x 4 + 2x 5 + x 6
Tamari lattice Counting intervals
Decomposition Smaller trees
B T (x) = x 3 + 2x 4 + 2x 5 + x 6
B T (1) = 6
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Decomposition Smaller trees
B T (x) = x 3 + 2x 4 + 2x 5 + x 6
Tamari lattice Counting intervals
Decomposition Smaller trees
B T (x) = x 3 + 2x 4 + 2x 5 + x 6
B T (1) = 6
Viviane Pons Intervals of the Tamari lattice
B T (x) = x 3 + 2x 4 + 2x 5 + x 6
B T (1) = 6
Tamari lattice Counting intervals
Decomposition Smaller trees
→
1 2
3
+
1 2
3
x 3 + x 2
→
1 2x 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
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Decomposition Smaller trees
→
1 2
3
+
1 2
3
x 3 + x 2
→
1 2x 2
1 2
3 4
5 6
x 3 .x.x 2
+ x .x.x + x .x + x .x.x + x .x.x + x .x
Tamari lattice Counting intervals
Decomposition Smaller trees
→
1 2
3
+
1 2
3
x 3 + x 2
→
1 2x 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
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Decomposition Smaller trees
→
1 2
3
+
1 2
3
x 3 + x 2
→
1 2x 2
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 .x.x + x .x.x + x .x
Tamari lattice Counting intervals
Decomposition Smaller trees
→
1 2
3
+
1 2
3
x 3 + x 2
→
1 2x 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
Viviane Pons Intervals of the Tamari lattice
Tamari lattice Counting intervals
Decomposition Smaller trees
→
1 2
3
+
1 2
3
x 3 + x 2
→
1 2x 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
x 3 .x.x 2 + x 3 .x.x + x 3 .x + x 2 .x.x 2 + x 2 .x.x
+ x .x
Tamari lattice Counting intervals
Decomposition Smaller trees
→
1 2
3
+
1 2
3
x 3 + x 2
→
1 2x 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
Viviane Pons Intervals of the Tamari lattice
→
1 2
3
+
1 2
3
x 3 + x 2
→
1 2x 2
(x 3 + x 2 ).x.(x 2 + x + 1) =
Tamari lattice Counting intervals
Decomposition Smaller trees
→
1 2
3
+
1 2
3
x 3 + x 2
→
1 2x 2
(x 3 + x 2 ).x.(x 2 + x + 1) = x B L (x) B R (x) − B R (1) x − 1
Viviane Pons Intervals of the Tamari lattice