We give a characterization of permutation polynomials over a finite field based on their coefficients, similar to Hermite's Criterion. Then, we use this result to obtain a formula for the total number of monic permutation binomials of degree less than 4p over F4p+1, where p and 4p+1 are primes, in terms of the numbers of three special types of permutation binomials. We also briefly discuss the case q=2p+1 with p and q primes.

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Journal Electronic Journal of Combinatorics
Masuda, A., Panario, D, & Wang, Q. (2006). The number of permutation binomials over double struck F sign 4p+1 where p and 4p + 1 are primes. Electronic Journal of Combinatorics, 13(1 R), 1–15.