Problem:
Score: 400 points

Problem Statement

Amidakuji is a traditional method of lottery in Japan.

To make an amidakuji, we first draw W parallel vertical lines, and then draw horizontal lines that connect them. The length of each vertical line is H+1 [cm], and the endpoints of the horizontal lines must be at 1, 2, 3, ..., or H [cm] from the top of a vertical line.

A valid amidakuji is an amidakuji that satisfies the following conditions:

No two horizontal lines share an endpoint.

The two endpoints of each horizontal lines must be at the same height.

A horizontal line must connect adjacent vertical lines.

Find the number of the valid amidakuji that satisfy the following condition, modulo 1\ 000\ 000\ 007: if we trace the path from the top of the leftmost vertical line to the bottom, always following horizontal lines when we encounter them, we reach the bottom of the K-th vertical line from the left.

For example, in the following amidakuji, we will reach the bottom of the fourth vertical line from the left.

Constraints

H is an integer between 1 and 100 (inclusive).

W is an integer between 1 and 8 (inclusive).

K is an integer between 1 and W (inclusive).

Input

Input is given from Standard Input in the following format:

H W K

Output

Print the number of the amidakuji that satisfy the condition, modulo 1\ 000\ 000\ 007.

Sample Input 1

1 3 2

Sample Output 1

1

Only the following one amidakuji satisfies the condition:

Sample Input 2

1 3 1

Sample Output 2

2

Only the following two amidakuji satisfy the condition:

Sample Input 3

2 3 3

Sample Output 3

1

Only the following one amidakuji satisfies the condition:

Sample Input 4

2 3 1

Sample Output 4

5

Only the following five amidakuji satisfy the condition:

Sample Input 5

7 1 1

Sample Output 5

1

As there is only one vertical line, we cannot draw any horizontal lines. Thus, there is only one amidakuji that satisfies the condition: the amidakuji with no horizontal lines.

Sample Input 6

15 8 5

Sample Output 6

437760187

Be sure to print the answer modulo 1\ 000\ 000\ 007.

Solution in Rust (1.15.1) (judge verdict: Accepted)

#[allow(dead_code)]
pub const M: u64 = 1_000_000_007;

fn main() {
    use std::io::Read;
    let mut buf = String::new();
    std::io::stdin().read_to_string(&mut buf).unwrap();
    let mut iter = buf.split_whitespace();
    
    let h: usize = iter.next().unwrap().parse().unwrap();
    let w: usize = iter.next().unwrap().parse().unwrap();
    let k: usize = iter.next().unwrap().parse::<usize>().unwrap() - 1;
    
    
    // dp[x][k] = x-1までの高さでkに出る通り
    let mut dp = vec![vec![0;w];h+1];
    dp[0][0] = 1u64;
    for y in 0..h {
        for ptn in 0..(1<<(w-1)) {
            let mut incorrect = false;
            if w >= 2 {
                for k in 0..w-2 {
                    if ((ptn >> k) & 1) == 1 && ((ptn >> (k+1)) & 1) == 1 {
                        incorrect = true;
                        break;
                    }
                }
            }
            if incorrect {continue;}

            for x in 0..w {
                let mut x2 = x;
                if ((ptn >> x) & 1) == 1 {
                    x2 += 1;
                } else if x>=1 && ((ptn >> (x-1)) & 1) == 1 {
                    x2 -= 1;
                }
                dp[y+1][x2] += dp[y][x];
                dp[y+1][x2] %= M;
            }
        }
    }
    println!("{}", dp[h][k]);
}