ON SUB-PERFECT AND PERFECT PARTITIONS OF INTEGERS
A perfect partition of a natural number is one which contains one, and only one, partition of every lower number. A sub-perfect partition of a number is such that, taking its parts positively or negatively, it is possible to compose in only one manner every lower number. Our results include explicit enumeration formulas and some combinatorial identities. The purpose of the research is to give a simple proof recurrence relations that count the number of ordered factorizations of natural numbers. Using the method of generating functions, we obtain a simple proof of an explicit formula for calculating McMahon ordered factorizations of natural numbers, and some new recurrence relations when these numbers have only two prime factors. For such natural numbers we found the relationship between the number of ordered factorizations and exponential Bell numbers and Stirling numbers of the second kind.