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 Mobius function. Using this result, we deduce an asymptotic expression for the number of such systems.