On an inclusion-exclusion formula based on the reflection principle

The n-candidate ballot problem corresponding to the standard Young tableau has been solved recently by Zeilberger (Discrete Math. 44 (1983) 325–326) by using the reflection principle. In this paper, a refinement of Zeilberger's approach is provided in which the reflection principle is formulated through the symmetric group and an inclusion-exclusion formula for the counting problem is developed. This approach reveals the nature in which successive applications of the reflection principle work.