  We find a formula for the number of permutation polynomials of degree q-2
over a finite field Fq, which has q elements, in terms of the permanent of a
matrix. We write down an expression for the number of permutation polynomials
of degree q-2 over a finite field Fq, using the permanent of a matrix whose
entries are pth roots of unity and using this obtain a nontrivial bound for the
number. Finally, we provide a formula for the number of permutation polynomials
of degree d less than q-2.
