Certain Partitions of Sets of Intergers

A partition of positive integers is called similis parition. A pair () is a complementing pair if , where ℕ = ℕ ∪ {0}. When is a finite set, we show that similis partitions () are completely determined by complementing -pairs, where is the total number of prime factors of elements of . We charaterize all the complementing -pairs for any > 2. This generalizes the works of de Bruijn [3] and I. Niven [8], and answers a question posed by I. Niven [8]