  We provide enumerating results for partial knight's paths of a given size. We
prove algebraically that zigzag knight's paths of a given size ending on the
$x$-axis are enumerated by the generalized Catalan numbers, and we give a
constructive bijection with peakless Motzkin paths of a given length. After
enumerating partial knight's paths of a given length, we prove that zigzag
knight's paths of a given length ending on the $x$-axis are counted by the
Catalan numbers. Finally, we give a constructive bijection with Dyck paths of a
given length.
