Enumerating Permutation Polynomials I: Permutations with Non-Maximal Degree

Abstract

Let C be a conjugation class of permutations of a finite field F q. We consider the function N C ( q) defined as the number of permutations in C for which the associated permutation polynomial has degree <q−2. In 1969, Wells proved a formula for N [3]( q) where [ k] denotes the conjugation class of k-cycles. We will prove formulas for N [ k] ( q) where k=4,5,6 and for the classes of permutations of type [2 2],[3 2],[4 2],[3 3] and [2 2 2]. Finally in the case q=2 n , we will prove a formula for the classes of permutations which are product of 2-cycles.