Labeled Factorization of Integers

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, . . . , n − 1}. We also give a new combinatorial interpretation of a class of generalized Stirling numbers. 1 Ordered and labeled factorization An ordered factorization of a positive integer n is a representation of n as an ordered product of integers, each factor greater than 1. The set of ordered factorizations of n will be denoted by F (n), and |F (n)| = f(n). For example, F (6) = {6, 2.3, 3.2}. So f(6) = 3. Every integer n > 1 has a canonical factorization into prime numbers p1, p2, . . ., namely n = p1 1 p m2 2 . . . p mr r , p1 < p2 < · · · < pr, mi > 0, 1 ≤ i ≤ r. (1) The enumeration function f(n) does not depend on the size of n but on the exponents mi. In particular we define Ω(n) = m1 +m2 + · · ·+mr, Ω(1) = 0. Note that the form of (1) may sometimes suggest a formula for f(n). For instance, • n = p gives f(n) = 2, the number of compositions of m. the electronic journal of combinatorics 16 (2009), #R50 1 • n = p1p2 . . . pr gives f(n) = r ∑ k=1 k!S(r, k), the r ordered Bell number; S(n, k) is the Stirling number of the second kind. A general formula for the number f(n, k) of ordered k-factorizations of n was found in 1893 by MacMahon [9] (also [1, p. 59]):