2025/09/23 by Sen‐Peng Eu, Eu, Sen-Peng, Tung-Shan Fu +3
Mathematics · Physics and Astronomy · #05A19 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2509.18981
openalex publication_date 2025/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a statistic traj on the ordered pairs (P,Q) of Dyck paths of size n, which counts the number of billiard trajectories in the grid polygon enclosed by P and -Q, where -Q is the path obtained by reflecting Q over the ground line. It turns out to coincide with the component statistic of meanders. In terms of grid polygon, we establish an involution on the set of such ordered pairs (P,Q) which either increases or decreases traj(P,Q) by 1. This proves a result by Di Francesco--Golinelli--Guitter that the numbers of semimeanders (meanders, respectively) of order n with even and odd numbers of components are equal if n is even and differ by a Catalan number (the square of a Catalan number, respectively) if n is odd. Some results about (-1)-evaluation of the generating functions for the statistic traj on restricted sets of Dyck paths are also presented.