Meanders and Dyck-Path Billiards

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. 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.