Natural exact covering systems and the reversion of the M��bius 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 $\sum_{k \ge 1} ��(k) x^k$, where $��(k)$ is the usual number-theoretic M��bius function. Using this result, we deduce an asymptotic expression for the number of such systems.