Permutations and cycles (1)

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 $O(100^3 + c)$ .

Problem information

Author: Unknown
Translator: Salvador Roura

Generation: 2026-01-25T11:19:11.851Z