A bijection for ordered factorizations
An ordered factorization of n is a representation of n as a product of integers \(\geq 2\); two such factorizations are considered the same only if they agree in both factors and order. Let \(\Psi\) (n) denote the class of all ordered factorizations \(\psi\) of n; and write \(k=k(\psi)\) for the number of factors in \(\psi\). The authors give two short proofs of the result: \(\sum_{\psi \in \Psi (n)}(-1)^{k(\psi)}=\mu (n)\) where \(\mu\) is the Möbius function. The first proof uses generating functions; the second exhibits an explicit bijection.
scientific article; zbMATH DE number 4150156