Natural exact covering systems and the reversion of the Möbius 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 ∑ _k ≥ 1μ (k) x^k , where μ (k) is the usual number-theoretic Möbius function. Using this result, we deduce an asymptotic expression for the number of such systems.