On Reduced Unicellular Hypermonopoles

Robert Cori Labri, Université Bordeaux 1 33405 Talence Cedex France

                     cori@labri.fr

Gábor Hetyei Department of Mathematics and Statistics UNC Charlotte Charlotte NC 28223-0001 USA

                 ghetyei@charlotte.edu

Abstract

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 cyclic 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.

1 Introduction

In the study of the combinatorics of the symmetric group many authors have been interested in the statistics of cycle lengths of products of pairs of permutations. In this note we revisit the particular case of counting the cycles of the product of two cyclic permutations, or dually finding the number of decompositions of a given permutation as a product of two cyclic permutations.

Our renewed interest in this topic comes from the study of hypermaps. In a recent paper [?] we showed that the problem of drawing a hypermap may be reduced to considering