Problem Description

There are $ N $ people, each pointing at one other person (other than themselves). No one is left unpointed.

When each person reports after how many steps they return to themselves, the maximum value is $ K $ steps.

Let's count how many such pointing arrangements exist. When counting the pointing arrangements, people are distinguished from each other.

The number of pointing arrangements can be very large, so find the remainder when divided by $ 1,000,000,007(=10^{9}+7) $.

Input Format

The input is given from the standard input in the following format:

$ N $ $ K $

• The first line contains two integers $N(2\ \leq\ N\ \leq\ 1000)$ and $K(1\ \leq\ K\ \leq\ N)$ separated by a space.

Output Format

Output the number of pointing arrangements modulo $ 1,000,000,007 $ in one line. Ensure a newline at the end of the output.

Sample Input and Output

Sample Input #1

3 3

Sample Output #1

2

Sample Input #2

4 2

Sample Output #2

3

Sample Input #3

3 2

Sample Output #3

0

Sample Input #4

1000 1000

Sample Output #4

756641425

Notes/Hints

Partial Points

This problem has partial points.

• If you solve the dataset where $2\ \leq\ N\ \leq\ 8$, you get $ 10 $ points.
• If you solve all datasets, you get an additional $ 140 $ points. The total is $ 150 $ points.

Sample Explanation 1

When each person is numbered $ 1 $, $ 2 $, $ 3 $, there are two pointing arrangements: $ 1 $→$ 2 $→$ 3 $→$ 1 $ and $ 1 $→$ 3 $→$ 2 $→$ 1 $.

Sample Explanation 2

When each person is numbered $ 1 $, $ 2 $, $ 3 $, $ 4 $, there are three pointing arrangements: $ 1 $→$ 2 $→$ 1 $, $ 3 $→$ 4 $→$ 3 $; $ 1 $→$ 3 $→$ 1 $, $ 2 $→$ 4 $→$ 2 $; and $ 1 $→$ 4 $→$ 1 $, $ 2 $→$ 3 $→$ 2 $.

Sample Explanation 3

There may be no pointing arrangement that satisfies the conditions.

Sample Explanation 4

The answer can be very large, so output the remainder when divided by $ 1,000,000,007 $.

见题面 Input 部分

见题面 Output 部分