Enumeration of Łukasiewicz paths modulo some patterns

Abstract

For any pattern $\alpha$ of length at most two, we enumerate equivalence classes of Łukasiewicz paths of length $n\ge 0$ where two paths are equivalent whenever the occurrence positions of $\alpha$ are identical on these paths. As a byproduct, we give a constructive bijection between Motzkin paths and some equivalence classes of Łukasiewicz paths.