Permutations and cycles (2) P64069

Write a program to count the number of permutations of $\{1, \ldots, n\}$ with exactly $k$ cycles, where $1 \le k \le n$.

For instance, of the six permutations of $\{1, 2, 3\}$, we have:

• two with one cycle, which are: $(2, 3, 1)$ and $(3, 1, 2)$.

• three with two cycles, which are: $(2, 1, 3)$, $(1, 3, 2)$ and $(3, 2, 1)$.

• one with three cycles, which is: $(1, 2, 3)$.

Input

Input consists of several cases, each with $n$ and $k$, such that $1 \le k \le n \le 1000$.

Output

For every case, count the number of permutations of $\{1, \ldots, n\}$ with $k$ cycles. As the result can be very large, make the computations modulo $10^8 + 7$.

Observation

Let $c$ be the number of cases. The expected solution has total cost $O(1000^2 + c)$. You can get up to 80 points with test cases where $n \le 100$, with a solution with cost $O(100^3 + c)$.