Natural exact covering systems and the reversion of the Möbius series

Abstract

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 ${\sum }_{k\ge 1}\mu (k)x^k$, where $\mu (k)$ is the usual number-theoretic Möbius function. Using this result, we deduce an asymptotic expression for the number of such systems.