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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,


Why is this true?

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up vote 2 down vote accepted

For the first see Ferrers diagram in Wikipedia's article on partition function. For the second, the first term on the right is the number of ways to write $n$ as a sum of $k$ plus terms smaller than $k$, as if you reduce the $k$ term by 1 you have a way to write $n-1$ as terms with largest term $k-1$. The second term is the number of ways to write $n$ as a sum of terms with at least two terms $k$ and nothing larger.

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