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arXiv · 1711.04109

Natural exact covering systems and the reversion of the M\"obius series

I. P. Goulden·Andrew Granville·L. Bruce Richmond·Jeffrey Shallit
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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 M\"obius function. Using this result, we deduce an asymptotic expression for the number of such systems.

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I. P. Goulden, Andrew Granville, L. Bruce Richmond, Jeffrey Shallit. 2017-11-11. Natural exact covering systems and the reversion of the M\"obius series. https://arxiv.org/abs/1711.04109

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