  In this note we observe that a bijection related to Littelmann's root
operators (for type $A_1$) transparently explains the well known enumeration by
length of walks on $\N$ (left factors of Dyck paths), as well as some other
enumerative coincidences. We indicate a relation with bijective solutions of
Bertrand's ballot problem: those can be mechanically transformed into bijective
proofs of the mentioned enumeration formula.
