Enumeration of Lattice paths with infinite types of steps and the Chung-Feller property

Abstract

In this paper we consider the lattice paths that go from the origin to a point on the right half of the plane with step set S={Ui=(1+i,1−i)|i≥0}⋃{Vi=(i,−i)|i≥1} such that the step Ui is assigned with weight ui and the step Vi is assigned with weight vi for i≥1 except that the step U0=(1,1) is assigned with weight 1. Let G(n,k) be the set of all lattice paths ending at the point (n,2k−n) and Gn,k=|G(n,k)|, and let U(n,k) (resp. V(n,k)) be the set of all lattice paths (resp. nonnegative lattice paths) ending at (2n−k,k) and Un,k=|U(n,k)| (resp. Vn,k=|V(n,k)|). We will show that (Gn,k)n,k∈N, (Un,k)n,k∈N and (Vn,k)n,k∈N are all Riordan arrays. Correlations between these Riordan arrays are studied. Consequently, a new Chung-Feller type property is obtained, and the bijective proof is provided. We also list numerous interesting examples.