ar X iv : 0 80 5 . 12 73 v 1 [ m at h . C O ] 9 M ay 2 00 8 BELL POLYNOMIALS AND k - GENERALIZED DYCK PATHS

Abstract. A k-generalized Dyck path of length n is a lattice path from (0, 0) to (n, 0) in the plane integer lattice Z × Z 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.