Enumeration problems associated with Donaghey’s transformation

In this paper we consider enumeration problems associated with Donaghey’s transformation. We discuss two groups of questions. The first one is related to the enumeration of fragments of transformation orbits, which are referred to as the “arcs”. The second group of questions is concerned with finding the number of vertices in rotation graphs — a specific family of graphs that is by nature an approximation of Donaghey’s transformation. The basic results of this work are formulated in the form of generating functions and corresponding asymptotics.