Riordan arrays, Łukasiewicz paths and Narayana polynomials

Abstract

By counting weighted Łukasiewicz paths and weighted Schröder paths, we obtain two Riordan arrays in which the bivariate Narayana polynomials occur in their first columns. We present a bijection between the weighted Łukasiewicz paths and the weighted Schröder paths, hence we obtain a new combinatorial interpretation for Schröder numbers: the nth Schröder number equals the number of Łukasiewicz paths of length n in which all up steps are weighted by 1, and other steps are weighted by 2. Furthermore, we provide some identities on the bivariate Narayana polynomials by considering two Riordan arrays related to weighted Motzkin paths.