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.