Séminaire Lotharingien de Combinatoire, 91B.34 (2024), 12 pp.
Guillaume Chapuy
On The Scaling of Random Tamari Intervals and Schnyder Woods of Random Triangulations (with an Asymptotic D-Finite Trick)
Abstract.
We consider a Tamari interval of size n (i.e., a pair
of Tamari-comparable Dyck paths) chosen uniformly at
random. We show that the typical height of points on the paths
scales as n3/4, with an explicit limit law.
By the Bernardi-Bonichon bijection,
this also applies to
Schnyder trees of random plane triangulations.
The exact solution of the model is based on polynomial equations with one and two catalytic variables.
To deduce convergence in law, we use a simple analytic method based on D-finiteness, which is essentially automatic.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
The following versions are available: