The hitting time subgroup, Łukasiewicz paths and Faber polynomials

The hitting time subgroup is a subgroup of the Riordan group and it came up recently in connection with Faber polynomials as studied in classical complex analysis. It turns out that the hitting time subgroup also arises in situations involving random walks and often is the distinguished leaf generating function for classes of ordered trees. These connections are the main focus of this paper.