Natural exact covering systems and the reversion of the Möbius series

2017/11/11 by I. P. Goulden, Goulden, I. P., Andrew Granville +5
Computer Science · Mathematics · #05A15 #05A16 #11A07 #11A25 #11N37 #68R15 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1711.04109

openalex publication_date 2017/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∑_{k ≥ 1} μ(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.

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