A Reflection Principle for Three Vicious Walkers

TL;DR

This paper introduces a reflection principle for three vicious lattice walkers, simplifying their enumeration by relating it to two-walker configurations and providing combinatorial insights into their generating functions.

Contribution

It develops a novel reflection principle for three vicious walkers, enabling reduction to two-walker problems and connecting to classical combinatorial models.

Findings

Established a reflection principle for three lattice walkers.

Reduced three-walker enumeration to two-walker configurations.

Provided a combinatorial interpretation of the generating function.

Abstract

We establish a reflection principle for three lattice walkers and use this principle to reduce the enumeration of the configurations of three vicious walkers to that of configurations of two vicious walkers. In the combinatorial treatment of two vicious walkers, we make connections to two-chain watermelons and to the classical ballot problem. Precisely, the reflection principle leads to a bijection between three walks $(L_1, L_2, L_3)$ such that $L_2$ intersects both $L_1$ and $L_3$ and three walks $(L_1, L_2, L_3)$ such that $L_1$ intersects $L_3$. Hence we find a combinatorial interpretation of the formula for the generating function for the number of configurations of three vicious walkers, originally derived by Bousquet-M\'elou by using the kernel method, and independently by Gessel by using tableaux and symmetric functions.