Two bijections on Tamari intervals
F. Chapoton, G. Chatel, V. Pons
Universit´e Claude Bernard Lyon
Universit´e Paris-Est Marne-la-Vall´ee
Fakult¨at f¨ur Mathematik, Universit¨at Wien
Definitions
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
Interval-poset
[
4 1
3 2
,
1 2
4 3
] →[ 1
2 3
4, 1 2 4
3 ] → 1
2 3
4
Rules of interval-posets:
a < c and a C c =⇒ b C c, b ∈ [a, c].
a < c and c C a =⇒ b C a, b ∈ [a, c].
Flows of forest of rooted trees
x
x ≥ −1
y
x x+y
-1
-1
-1
1
1
1
3
3 4 4
-1
-1
2
2 1 0
3 3
-1
0
0
2
2
1 1
The exit rate of a forest is the sum of the exit rates of the trees.
A closed flow is a flow of a forest of rooted trees with exit rate 0.
Statistics on interval-posets
I =
1
3 2
5 4
6
trees(I) = 4
ir(I) = 1 J =
4 3
5 2
1
6 trees(J) = 1 ir(J) = 4
trees(I) = number of red trees of I, ir(I) = largest k s.t. there is no relation i C i + 1 for i ∈ [1, k].
Results Theorem
The number of closed flows of a given forest F is the number of elements smaller than or equal to a certain binary tree T (F ) in the Tamari order.
Bijective proof
-1
-1
1
1
1
0
0 1 1
-1
-1
2
2 1 0
0
-1
0
0
1
1
0
-1
-1
1
1
1
0
0 1 1
-1
-1
2
2 1 0
0
-1
0
0
1
1
0
-1
-1
1
1
1
0
0 1 1
-1
-1
2
2 1 0
0
0
0
0
0
0
0
1 2
3 4 5
6 7 8
9 10 11
1 2
3 4 5
6 7 8
9 10 11
1 2
3 4 5
6 7 8
9 10 11
-1
-1
1
1
1
0
0 1 1
-1
0
1
1 1 0
0
0
0
0
0
0
0
-1
-1
1
1
1
0
0 1 1
0
0
0
0 0 0
0
0
0
0
0
0
0
-1
0
0
0
1
0
0 1 1
0
0
0
0 0 0
0
0
0
0
0
0
0
1 2
3 4 5
6 7 8
9 10 11
1 2
3 4 5
6 7 8
9 10 11
1 2
3 4 5
6 7 8
9 10 11
Comments
How statistics on flows can be read on the corresponding interval-poset ? The flows appear in the study of the pre-Lie operad (see [2]) which doesn’t
seem related to the Tamari order. Can we provide an explanation of this link ? Each open flow can be sent to a unique closed flow. What is the connection with the Tamari order ?
Theorem
Let I be an interval-poset of size n such that trees(I) = x and ir(I) = y.
There exists another interval-poset J of size n such that trees(J) = y and ir(J) = x.
Bijective proof
I =
1
3 2
5 6
4
7
2 8
1 2
3
1 2 3
4
1
1 1
0
∅
1 2
13
1 1
J =
4 5
3
6
7 2
1 8
2
2
1 3
2
3 1
4
1
1 1
0
∅
2
1 3
1
1 1
trees(I) = 4 = ir(J) ir(I) = 1 = trees(J)
Comments
This bijection yields a non trivial equality between two functionnal equations. Can we prove it algebrically ?
Citations
M. Bousquet-M´elou, E. Fusy, and L.-F. Pr´eville-Ratelle.
The number of intervals in the m-Tamari lattices.
Electron. J. Combin., 18(2):Paper 31, 26, 2011.
F. Chapoton.
Flows on rooted trees and the Menous-Novelli-Thibon idempotents.
arXiv:1203.1780, 2013.
Fr´ed´eric Chapoton, Gr´egory Chatel, Viviane Pons Universit´e Paris-Est Marne-la-Vall´ee
Two bijections on Tamari intervals
