Permutations and cycles (2) P64069

Write a program to count the number of permutations of {1, …, n} with exactly k cycles, where 1 ≤ k ≤ 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 ≤ k ≤ n ≤ 1000.

Output

For every case, count the number of permutations of {1, …, n} with k cycles. As the result can be very large, make the computations modulo 10⁸ + 7.

Observation

Let c be the number of cases. The expected solution has total cost O(1000² + c). You can get up to 80 points with test cases where n ≤ 100, with a solution with cost O(100³ + c).