Relatively Prime Uniform Partitions

A partition of a positive integer n is a sequence of non increasing positive integers say λ = (λ1, · · · , λ1(f1 times), · · · , λk, · · · , λk (fk times)), with λi > λi+1, whose sum equals n. If gcd(λ1, · · · , λk) = 1, we say that λ is a relatively prime partition. If fi = 1 ∀i = 1, 2, · · · , k, we say that λ is a distinct partition. If fi = fj ∀i 6= j, then we say that λ is an uniform partition. In this note, the class of partitions which are both uniform and relatively prime are analyzed. They are found to have connections to distinct partitions. Furthermore, we term a partition of a certain kind as Conjugate Closed (abbreviated CC) if its conjugate is also of the same kind. Enumeration of CCuniform partitions, CC-relatively prime partitions, and CC-uniform relatively prime partitions are studied; these enumeration formulas involves functions from multiplicative number theory such as divisors counting function and Lioville’s function.