Natural exact covering systems and the reversion of the Mobius series

We prove that the number of natural exact covering systems of cardinality k is equal to the coefficient of x(k) in the reversion of the power series Sigma(k >= 1) mu(k)x(k), where mu(k) is the usual number-theoretic Mobius function. Using this result, we deduce an asymptotic expression for the number of such systems.