Recent news piqued my interest in integer partitions again. I'm working my way back through an old text and I'm completely hung up on this problem:
Recall that $p_k(n)$ is the number of partitions of $n$ into exactly $k$ parts. Prove that for all positive integers $k\leq n$, the inequality $p_k(n) \leq (n-k+1)^{k-1}$ holds.