Bell Polynomials and $k$-generalized Dyck Paths

Abstract

A {\em k-generalized Dyck path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $\mathbb{Z}\times \mathbb{Z}$ consisting of horizontal-steps $(k,0)$ for a given integer $k\ge 0$, 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.

15pages, 1 figure. To appear in Discrete Applied Mathematics

Cited by in corpus (9)

• Lagrange Inversion
• The Wigner distribution function for the $\mathfrak{su}(2)$ finite oscillator and Dyck paths
• Bijections on $r$-Shi and $r$-Catalan Arrangements
• Multi-colored dimer models in one-dimension: lattice paths and generalized Rogers--Ramanujan identities
• Lattice paths enumerations weighted by ascent lengths
• Redundant generating functions in lattice path enumeration
• Potential Polynomials and Motzkin Paths
• A q-analogue of some binomial coefficient identities of Y. Sun
• A simple bijection between binary trees and colored ternary trees