  Using a noncommutative analog of Chevalley's decomposition of polynomials
into symmetric polynomials times coinvariants due to Bergeron, Reutenauer,
Rosas, and Zabrocki we compute the graded Frobenius series for their two sets
of noncommutative harmonics with respect to the left action of the symmetric
group (acting on variables). We use these results to derive the Frobenius
series for the enveloping algebra of the derived free Lie algebra in n
variables.
