On the Number of Partitions of n into Exactly m Parts Whose Even Parts are Distinct

Let $ped(n)$ be the number of partitions of n whose even parts are distinct and whose odd parts are unrestricted. For a positive integer m, let $ped(n, m)$ be the number of all possible partitions of the number n into exactly m parts whose even parts are distinct and whose odd parts are unrestricted. In this paper, we give new recurrence formulas for $ped(n,m)$ as well as explicit formulas for $ped(n, m)$, when $m=2, 3$ and $m=4$. For a positive integer q and j∈\0,1,2,…,q-1\, we also give a recurrence formula for pq,j(n,m) the number of partitions of n into m parts such that the parts congruent to $-j$ modulo q are distinct, where other parts are unrestricted.