Labeled Factorization of Integers

摘要

The labeled factorizations of a positive integer n are obtained as a completion ofthe set of ordered factorizations of n. This follows a new technique for generatingordered factorizations found by extending a method for unordered factorizationsthat relies on partitioning the multiset of prime factors of n. Our results includeexplicit enumeration formulas and some combinatorial identities. It is proved thatlabeled factorizations of n are equinumerous with the systems of complementingsubsets of {0,1, ,n — 1}. We also give a new combinatorial interpretation of aclass of generalized Stirling numbers.