Generalized Delannoy paths with cyclically shifting boundaries

In this paper, we consider the generalized Delannoy paths with steps N_i = (0, j) , j ≥ 1 and H_i = (1, j) , j ≥ 0 , where all steps are weighted by u_i for N_i and v_j ( v_0 = 1 ) for H_j . 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, an,k) with m -flaws, where P is a generalized Delannoy path (with steps N_i and H_0 ) from (0,0) to (n,k) , an,k is a k -part composition of n , and a flaw is a horizontal step (1,0) of P below the boundary ∂ an,k .