Enumerating permutation polynomials over finite fields by degree. II

This note is a continuation of a paper by the same authors that appeared in 2002 in the same journal. First we extend the method of the previous paper proving an asymptotic formula for the number of permutations for which the associated permutation polynomial has d coefficients in specified fixed positions equal to 0. This also applies to the function Nq,d that counts the number of permutations for which the associated permutation polynomial has degree <q-d-1. Next we adopt a more precise approach to show that the asymptotic formula Nq,d∼q!/qd holds for d⩽αq and α=0.03983.