Bijections for Rota–Baxter words and Schröder paths
Rota–Baxter words were first introduced by Ebrahimi-Fard and Guo as canonical bases of free Rota–Baxter algebras. In this paper, we find that the Rota–Baxter words $R(n,n)$ with one idempotent generator and one idempotent operator are counted by the Catalan number. Then we construct some bijections between Rota–Baxter words and Schröder paths. In particular, we consider enumeration and generating functions of bracketed Rota–Baxter words and Rota–Baxter words whose right brackets are behind the rightmost left bracket, by using the Schröder paths viewpoint.