Bijections for Rota–Baxter words and Schröder paths

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