Quoting from Wolfram MathWorld, "$P(n,k)$ denotes the number of ways of writing $n$ as a sum of exactly $k$ terms or, equivalently, the number of partitions into parts of which the largest is exactly $k$."
Why are these two definitions equivalent? What is the mapping?
The page also states a recurrence,
$$P(n,k)=P(n-1,k-1)+P(n-k,k)$$
Why is this true?