Some simple bijections involving lattice walks and ballot sequences

Marc van Leeuwen (1)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]

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.

Mots clés: bijective proof; lattice path; ballot sequence; raising operator

Domaines: Combinatoire [math.CO]

Marc van Leeuwen. Some simple bijections involving lattice walks and ballot sequences. 2010. ⟨hal-00528787⟩