Enumeration of partial Łukasiewicz paths

Jean-Luc Baril (1) , Helmut Prodinger

1 LIB - Laboratoire d'Informatique de Bourgogne [Dijon]

Lukasiewicz paths are lattice paths in N 2 starting at the origin, ending on the x-axis, and consisting of steps in the set {(1, k), k ≥ −1}. We give bivariate generating functions and exact values for the number of n-length prefixes (resp. suffixes) of these paths ending (resp. starting) at height k ≥ 0 with a given type of step. We make a similar study for paths of bounded height, and we prove that the average height of n-length paths ending at a fixed height behaves as √ πn when n → ∞. Finally, we study prefixes of alternate Lukasiewicz paths, i.e., Lukasiewicz paths that do not contain two consecutive steps in the same direction.

Mots clés: Asymptotics; Enumeration

Domaines: Combinatoire [math.CO]

Jean-Luc Baril, Helmut Prodinger. Enumeration of partial Łukasiewicz paths. Enumerative combinatorics and applications, 2022, 3, ⟨10.54550/eca2023v3s1r2⟩. ⟨hal-03896485⟩