Hi Codeforces community,

Recently I have come across a problem which turned out to be tough for me. I hope that I can get some help from you.

Problem Statement

A permutation A of first N integers from 1 to N is good if it has exactly K good positions. A position i is good only if abs(A[i] - i) = 1. The task is to count how many permutation of first N integers like that, modulo 10⁹ + 7.

Input

N and K, 1 ≤ N ≤ 1000, 0 ≤ K ≤ N

Output

Number of permutation of first N integers from 1 to N that has exactly K good positions, modulo 10⁹ + 7

Example

For N = 3, K = 2, there are 4 permutations that has 2 good positions. They are (1, 3, 2) , (2, 1, 3) , (2, 3, 1) , (3, 1, 2).

You may want to submit your solution here (written in Vietnamese, required SPOJ account): http://www.spoj.com/PTIT/problems/P172PROI/

I think it is a DP problem although I could not come up with a solution or any idea. Any help will be appreciated.

Thanks in advance.

I saw this submission of problem 678A - Вася любит числа and tried to hack it with case n=1000000000 and k=1.

18417780

It passes within the problem's time limit 0.5s, but I thought it should be a TLE (I tested on ideone and it runs about 0.91s). I can't think of any reason but CF judge is really fast. Any other ideas?