Generalized Delannoy paths with cyclically shifting boundaries

Abstract

In this paper, we consider the generalized Delannoy paths with steps Ni = (0, i), i ≥ 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 Schro¨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 Ni and H0) 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 .