Weighted Dyck paths with special restrictions on the levels of valleys

TL;DR

This paper studies weighted Dyck paths with valley level restrictions, providing explicit generating functions, establishing connections to classical combinatorial structures, and constructing bijections under special weights.

Contribution

It introduces explicit formulas for the generating functions of these paths and links them to well-known combinatorial objects through bijections and special weights.

Findings

Derived explicit generating functions for weighted Dyck paths with restrictions.

Established connections between these paths and Motzkin, Schröder, Delannoy paths, and k-ary trees.

Constructed bijections under specific weight functions.

Abstract

This paper concentrates on the set $\mathcal{V}_n$ of weighted Dyck paths of length $2n$ with special restrictions on the level of valleys. We first give its explicit formula of the counting generating function in terms of certain weight functions. When the weight functions are specialized, some connections are builded between $\mathcal{V}_n$ and other classical combinatorial structures such as $(a,b)$-Motzkin paths, $q$-Schr\"{o}der paths, Delannoy paths and complete $k$-ary trees. Some bijections are also established between these settings and $\mathcal{V}_n$ subject to certain special weight functions.