Combinatorics of non-ambiguous trees

This article investigates combinatorial properties of non-ambiguous trees. These objects we define may be seen either as binary trees drawn on a grid with some constraints, or as a subset of the tree-like tableaux previously defined by Aval, Boussicault and Nadeau. The enumeration of non-ambiguous trees satisfying some additional constraints allows us to give elegant combinatorial proofs of identities due to Carlitz, and to Ehrenborg and Steingr\'imsson. We also provide a hook formula to count the number of non-ambiguous trees with a given underlying tree. Finally, we use non-ambiguous trees to describe a very natural bijection between parallelogram polyominoes and binary trees.

The number of non-ambiguous trees A136127 = (a k ) k≥1 = (1, 2, 5, 16, 63, 294, 1585, 9692, . . . ) The number a n of non-ambiguous trees of size n is equal to the number of permutations having all its excedences placed at the begining.

tree-like tableau
The ending points of two left edges can be in the same row; The x-coordinates of the ending points of two different edges are different; The x-coordinates of the ending points of the left edges form an interval [1, k].
These properties are the same for the right edges. Total order on the set of the left (resp. right) edges We get two total orders on the ending points of the left (resp. right) edges Adrien Boussicault (LaBRI) Combinatorics of non-ambiguous trees UQTR -6/5/2015 17 / 38 The non-ambiguous trees are represented by (α L , α R )   The constraints verifyed by the ending points of the left edges ( α L ) Combinatorics of non-ambiguous trees UQTR -6/5/2015 24 / 38 Non-ambiguous trees with a fixed underlying tree We associate each edge with an integer n e . Given a left (resp.right) edge e, the integer n e is the number of left (resp. right) edges contained in the subtree whose root is the ending point of e, plus 1. The compete non-ambiguous trees A complete non-ambiguous tree is a non-ambiguous tree whose vertices have only 0 or 2 children. size = nb of internal vertices = nb of left edges = nb of right edges = 6

Proof
The Bessel function of order 0 is the solution of the differential equation: such that the first coefficients in its series expansion are j 0 = 1 and j 1 = 0.
We try