Identities involving weighted Catalan, Schroder and Motzkin paths

TL;DR

This paper explores relationships and identities among weighted Catalan, Motzkin, and Schr"oder numbers and paths, providing combinatorial proofs and a bijection between specific Catalan path sets.

Contribution

It introduces new identities linking these weighted path numbers and offers combinatorial proofs and a novel bijection between Catalan path sets.

Findings

Derived new identities connecting weighted Catalan, Motzkin, and Schr"oder numbers.

Provided combinatorial proofs for the established identities.

Established a bijection between Catalan paths with valleys and those with N steps in even positions.

Abstract

In this paper, we investigate the weighted Catalan, Motzkin and Schr\"oder numbers together with the corresponding weighted paths. The relation between these numbers is illustrated by three equations, which also lead to some known and new interesting identities. To show these three equations, we provide combinatorial proofs. One byproduct is to find a bijection between two sets of Catalan paths: one consisting of those with $k$ valleys, and the other consisting of $k$ $\mathbf{N}$ steps in even positions.