A8905. Positions in Permutations

Permutation $p$ is an ordered set of integers $p_{1},p_{2},...,p_{n}$ , consisting of $n$ distinct positive integers, each of them doesn't exceed $n$ . We'll denote the $i$ -th element of permutation $p$ as $p_{i}$ . We'll call number $n$ the size or the length of permutation $p_{1},p_{2},...,p_{n}$ .

We'll call position $i$ ( $1<=i<=n$ ) in permutation $p_{1},p_{2},...,p_{n}$ good, if $|p[i]-i|=1$ . Count the number of permutations of size $n$ with exactly $k$ good positions. Print the answer modulo $1000000007$ ( $10^{9}+7$ ).

The single line contains two space-separated integers $n$ and $k$ ( $1<=n<=1000,0<=k<=n$ ).

Print the number of permutations of length $n$ with exactly $k$ good positions modulo $1000000007$ ( $10^{9}+7$ ).

The only permutation of size 1 has 0 good positions.

Permutation $(1,2)$ has 0 good positions, and permutation $(2,1)$ has 2 positions.

Permutations of size 3:

1. $(1,2,3)$ — 0 positions
2.  — 2 positions
3.  — 2 positions
4.  — 2 positions
5.  — 2 positions
6. $(3,2,1)$ — 0 positions