A bijection for ordered factorizations
type: Item
instance of: scholarly article
title: A bijection for ordered factorizations
zbMATH Open document ID: 0702.05002
DOI: 10.1016/0097-3165(90)90040-4
author: Richard Garfield
author: Donald E. Knuth
author: Herbert S. Wilf
published in: Journal of Combinatorial Theory. Series A
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 11A51
zbMATH DE Number: 4150156
zbMATH Keywords: ordered factorization
zbMATH Keywords: Möbius function
zbMATH Keywords: generating functions
MaRDI profile type: Publication
full work available at URL: 0097-3165(90)90040-4
OpenAlex ID: W2037618871
Wikidata QID: Q105602959
description: scientific article; zbMATH DE number 4150156