  The problem of counting unicellular hypermonopoles by the number of their
hyperedges is equivalent to describing the cycle length distribution of a
product of two circular permutations, first solved by Zagier. The solution of
this problem has also been used in the study of the cycle graph model of Bafna
and Pevzner and of related models in mathematical biology. In this paper we
develop a method to compute the finite number of reduced unicellular
hypermonopoles of a given genus. The problem of representing any hypermap as a
drawing is known to be simplifiable to solving the same problem for reduced
unicellular hypermonopoles. We also outline a correspondence between our
hypermap model, the cycle graph model of Bafna and Pevzner, and the polygon
gluing model of Alexeev and Zograf. Reduced unicellular hypermonopoles
correspond to reduced objects in the other models as well, and the notion of
genus is the same.
