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