Natural exact covering systems and the reversion of the Möbius series

11/11/2017

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by   I. P. Goulden, et al.

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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.