Bell Polynomials and $k$-generalized Dyck Paths

TL;DR

This paper explores the combinatorial properties of k-generalized Dyck paths, using Bell polynomials to derive generating functions and new identities, including novel formulas for Catalan numbers.

Contribution

It introduces a novel application of Bell polynomials to analyze k-generalized Dyck paths and derives new combinatorial identities and formulas for Catalan numbers.

Findings

Derived generating functions using Bell polynomials

Established new combinatorial identities involving Dyck paths

Produced two new formulas for Catalan numbers

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 $k\geq 0$, 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.