Q-generalization of a ballot problem

Abstract

n-dimensional lattice paths which do not touch the hyperplanes x i − x i+1 =⇔-1, i = 1,2,…, n − 1, and x n − x 1=−1− K are enumerated by certain statistics, one of which is MacMahon's major index, the others being variations of it. By a reflection-like proof, a formula involving determinants is obtained. It is a q-extension of Filaseta's (1985) expression for the number of elections in a specific ballot problem.