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

TL;DR

This paper establishes a novel connection between natural exact covering systems and the reversion of a power series involving the M"obius function, leading to an asymptotic count of these systems.

Contribution

It introduces a new relationship linking covering systems to power series reversion of the M"obius series, providing asymptotic estimates.

Findings

Number of natural exact covering systems of size k equals coefficient of x^k in the series reversion.

Derived an asymptotic formula for the count of such covering systems.

Connected combinatorial structures with analytic series reversion techniques.

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.