Generating functionology
Left-inversion of combinatorial sums
Proceedings of the 7th conference on Formal power series and algebraic combinatorics
European Journal of Combinatorics
Catalan, Motzkin, and Riordan numbers
Discrete Mathematics - Special issue on selected papers in honor of Henry W. Gould
Combinatorial Enumeration
General identities on Bell polynomials
Computers & Mathematics with Applications
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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.