Lattice Paths, Reflections, & Dimension-Changing Bijections

Abstract

We enumerate various families of planar lattice paths consisting of unit steps in directions \( {N}\), \({S}\), \({E}\), or \({W}\), which do not cross the \(x\)-axis or both \(x\)- and \(y\)-axes. The proofs are purely combinatorial throughout, using either reflections or bijections between these \({NSEW}\)-paths and linear \({NS}\)-paths. We also consider other dimension-changing bijections.