Counting smaller trees in the Tamari order
Gr´ egory Chatel, Viviane Pons
Laboratoire d’Informatique Gaspard-Monge Universit´e Paris-Est Marne-la-Vall´ee
5 bd Descartes – 77454 Marne-la-Vall´ee Cedex 2 – France
The right rotation on binary trees
x
y
A B
C →
x A y
B C
The right rotation is the cover relation of the Tamari order on binary trees.
The Tamari lattice
Main result
Given a binary tree T , we define its Tamari polynomial BT (x) by:
B∅ := 1
BT (x) := xBL(x)xBR(x) − BR(1) x − 1
with T =
L R
BT (x) counts the number of trees smaller than or equal to T in the Tamari order according to the number of nodes on their left border. In particular, BT (1) is the number of trees smaller than T .
Example
BT (x) = xBL(x)xBR(x)−Bx−1 R(1)
BT (x) = x(x2 + x3)(1 + x + x2) BT (x) = x3 + 2x4 + 2x5 + x6
BT (1) = 6
••
••
••
••
••
••
•
••
••
•
•
••
•
••
••
••
••
••
••
••
The Tamari lattice is a quotient of the weak order
Each binary tree has a unique labelling as binary search tree. The linear extensions of these labelled binary trees are intervals of the weak order:
the sylvester classes.
4 3 2 1
1234
4 3 1
2
2134
4 2
1 3
1324 3124
3 2 1
4
1243 1423 4123
4 1
3 2
2314
3 1
2
4
2143 2413 4213
2
1 4
3
1342 3142 3412
4 1
2 3
3214
1
4 3 2
2341
2 1 3
4
1432 4132 4312
1
4 2
3
3241
1 3 2 4
2431 4231
1 2
4 3
3421
1 2
3 4
4321
An interval of the Tamari lattice can be seen as a union of sylvester
classes. It is represented as a poset whose linear extensions correspond to these sylvester classes.
[
4 2
1 3
, 1 2 4
3
] → {1324, 3124, 1342, 3142,
3412, 1432, 4132, 4312}
→ 2
1 3
4
Tamari Interval-posets
An interval-poset is a poset representing an interval of the Tamari lattice. With each
binary tree T , one can associate an increasing and a decreasing forest. They correspond respectively to the initial interval [Tmin, T ] and the final interval [T , Tmax] of the
Tamari lattice.
[
4
2 5
1 3 9
7
6 8
, Tmax ] −→
1 2
3
4 5
6 7 9
8
[ Tmin ,
4 1
2 3
5 7
6 9
8
] −→ 4
1 2 3
5 7 6
9 8
To construct the interval-poset [T1, T2], we combine the decreasing forest of T1 and the increasing forest of T2.
[
4
2 5
1 3 9
7
6 8
,
4 1
2 3
5
7
6 9
8
] −→
4 1 2 3
5 7 9 6 8
The number of trees smaller than or equal to T is the number of intervals [T 0, T ] having T as maximal element. Our proof is based on a combinatorial interpretation of the
bilinear operation BT .
1
2 3
4
5
6
[ 3
1 2
, 3 1 2
] + [ 3 1 2
, 3 1 2
] → 3
1 2 + 3
1 2
[ 2
1 , 2
1 ]→ 2 1
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
3 1 2
6
5 +
4
3 1 2
6
5 +
4
3 1 2
6 5
[ , ]+[ , ]+[ , ]+[ , ]+[ , ]+[ , ] x6 + x5 + x4 + x5 + x4 + x3
Other results and perspectives
New proof of the number of intervals of the Tamari lattice, q-generalization of the main result, generalization to m-Tamari, relations with flows of rooted trees, . . .
Gr´egory Chatel and Viviane Pons Universit´e Paris-Est Marne-la-Vall´ee
Counting smaller trees in the Tamari order
