The number of irreducible polynomials and Lyndon words with given trace

The trace of a degree n polynomial f(x) over GF(q) is the coefficient of x(n-1). Carlitz [ Proc. Amer. Math. Soc., 3 (1952), pp. 693-700] obtained an expression I-q(n,t) for the number of monic irreducible polynomials over GF ( q) of degree n and trace t. Using a different approach, we derive a simple explicit expression for I-q(n,t). If t > 0, I-q(n,t) = (Sigma mu (d)q(n/d))/ (qn), where the sum is over all divisors d of n which are relatively prime to q. This same approach is used to count L-q(n,t), the number of q-ary Lyndon words whose characters sum to t mod q. This number is given by L-q(n,t) = (Sigma gcd(d,q)mu (d)q(n/d))/(qn), where the sum is over all divisors d of n for which gcd(d,q)\t. Both results rely on a new form of Mobius inversion.