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

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