Bell polynomials and kk-generalized Dyck paths

A k-generalized Dyck path   of length nn is a lattice path from (0,0)(0,0) to (n,0)(n,0) in the plane integer lattice Z×ZZ×Z consisting of horizontal-steps (k,0)(k,0) for a given integer k≥0k≥0, up-steps (1,1)(1,1), and down-steps (1,−1)(1,−1), which never passes below the xx-axis. The present paper studies three kinds of statistics on kk-generalized Dyck paths: “number of uu-segments”, “number of internal uu-segments” and “number of (u,h)(u,h)-segments”. The Lagrange inversion formula is used to represent the generating function for the number of kk-generalized Dyck paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. Many important special cases are considered leading to several surprising observations. Moreover, enumeration results related to uu-segments and (u,h)(u,h)-segments are also established, which produce many new combinatorial identities, and specially, two new expressions for Catalan numbers.