In this paper, we consider the generalized Delannoy paths with steps Nᵢ = (0, j), j 1 and Hᵢ = (1, j), j 0, where all steps are weighted by uᵢ for Nᵢ and vⱼ (v₀ = 1) for Hⱼ. By the Riordan array theory, we provide a counting formula for the number of all generalized Delannoy paths dominated by a cyclically shifting piecewise linear boundary of varying slopes. Our main result can be viewed as a unified generalization of the well-known enumerative formulas for the generalized Dyck and Schr? der paths from (0, 0) to (kn, n) staying above the line x = ky. We also study the number of generalized Delannoy path boundary pairs (P, a₍, ₊) with m -flaws, where P is a generalized Delannoy path (with steps Nᵢ and H₀) from (0, 0) to (n, k), a₍, ₊ is a k -part composition of n, and a flaw is a horizontal step (1, 0) of P below the boundary a₍, ₊.
Zhang et al. (Wed,) studied this question.