Work

Meanders and Dyck-Path Billiards

Year: 2025

Type: article

Abstract: 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.

Source: RAIRO - Theoretical Informatics and Applications

DOI: 10.1051/ita/2025002

Authors: Sen-Peng Eu, Tung-Shan Fu, Hsiang-Chun Hsu

Institutions: National Taiwan Normal University, People's Liberation Army Air Force, National Pingtung University, Tamkang University

Language: English

Topic: Advanced Combinatorial Mathematics

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid:

Funder: National Science and Technology Council