The Number of Permutation Polynomials of a Given Degree Over a Finite Field

We relate the number of permutation polynomials in Fq[x] of degree d≤q−2 to the solutions (x1,x2,…,xq) of a system of linear equations over Fq, with the added restriction that xi≠0 and xi≠xj whenever i≠j. Using this we find an expression for the number of permutation polynomials of degree p−2 in Fp[x] in terms of the permanent of a Vandermonde matrix whose entries are the primitive pth roots of unity. This leads to nontrivial bounds for the number of such permutation polynomials. We provide numerical examples to illustrate our method and indicate how our results can be generalised to polynomials of other degrees.