Partitions into an exact number of parts

Abstract In this paper, we examine the functions $$P_k(n)$$ P k ( n ) , which counts the partitions of n into exactly k parts, and $$Q_k(n)$$ Q k ( n ) , which counts the partitions of n into exactly k distinct parts. These partition functions are closely linked to two classical identities of Euler. We explore this connection and establish several new relationships between $$P_k(n)$$ P k ( n ) and $$Q_k(n)$$ Q k ( n ) . We give both analytic and combinatorial proofs of the theorems.