Bell polynomials and k-generalized Dyck paths

A k-generalized Dyck path of length n is a lattice path from (0,0) to (n,0) in the plane integer lattice ZxZ consisting of horizontal-steps (k,0) for a given integer k>=0, up-steps (1,1), and down-steps (1,-1), which never passes below the x-axis. The present paper studies three kinds of statistics on k-generalized Dyck paths: ''number of u-segments'', ''number of internal u-segments'' and ''number of (u,h)-segments''. The Lagrange inversion formula is used to represent the generating function for the number of k-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 u-segments and (u,h)-segments are also established, which produce many new combinatorial identities, and specially, two new expressions for Catalan numbers.