P1025 [NOIP 2001 Senior] Integer Partition

Partition the integer $n$ into $k$ non-empty parts, and regard two partitions as the same if they differ only in order. For example, when $n=7$, $k=3$, the following three partitions are considered the same: $1, 1, 5$; $1, 5, 1$; $5, 1, 1$. How many different partitions are there?

Two integers $n$ and $k$. Constraints: $6 < n \le 200$, $2 \le k \le 6$.

One integer: the number of different partitions.

There are four partitions when $n=7$, $k=3$: $1, 1, 5$; $1, 2, 4$; $1, 3, 3$; $2, 2, 3$. Problem Source: NOIP 2001 Senior, Problem 2. Translated by ChatGPT 5