Labeled Factorization of Integers

Abstract

The labeled factorizations of a positive integer $n$ are obtained as a completion of the set of ordered factorizations of $n$. This follows a new technique for generating ordered factorizations found by extending a method for unordered factorizations that relies on partitioning the multiset of prime factors of $n$. Our results include explicit enumeration formulas and some combinatorial identities. It is proved that labeled factorizations of $n$ are equinumerous with the systems of complementing subsets of $\{0,1,\dots,n-1\}$. We also give a new combinatorial interpretation of a class of generalized Stirling numbers.