A Reflection Principle for Three Vicious Walkers
نویسندگان
چکیده
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 (L1, L2, L3) such that L2 intersects both L1 and L3 and three walks (L1, L2, L3) such that L1 intersects L3. 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-Mélou by using the kernel method, and independently by Gessel by using tableaux and symmetric functions.
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