Average Cayley Genus for Cayley Maps of Dihedral Groups Generated by Their Reflections

Let \(\Gamma\) be a finite group and let \(\Delta\) be a generating set for \(\Gamma\). A Cayley map associated with \(\Gamma\) and \(\Delta\) is an oriented 2-cell embedding of the Cayley graph \(G_\Delta(\Gamma)\) such that the rotation of arcs emanating from each vertex is determined by a unique cyclic permutation of generators and their inverses. A formula for the average Cayley genus is known for the dihedral group with generating set consisting of all the reflections. However, the known formula involves sums of certain coefficients of a generating function and its format does not specifically indicate the Cayley genus distribution. We determine a simplified formula for this average Cayley genus as well as provide improved understanding of the Cayley genus distribution.