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)

use std::io::prelude::*;

fn input<T>() -> T
where
    T: std::str::FromStr,
{
    let stdin = std::io::stdin();
    let token = stdin
        .lock()
        .bytes()
        .map(|c| c.unwrap() as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect::<String>();
    token.parse().ok().unwrap()
}

fn main() {
    let h = input::<usize>();
    let w = input::<usize>();
    let k = input::<usize>();
    const MOD: usize = 1000000007;

    let mut dp = vec![vec![0; w as usize + 1]; h as usize + 1];
    dp[0][1] = 1;
    for i in 0..h {
        for j in 1..w + 1 {
            for k in 0..1 << (w - 1) {
                if w > 1 {
                    let mut ok = true;
                    for l in 0..w - 2 {
                        if k & (1 << l) > 0 && k & (1 << (l + 1)) > 0 {
                            ok = false;
                        }
                    }
                    if !ok {
                        continue;
                    }
                }

                if j > 1 && k & (1 << (j - 2)) > 0 {
                    dp[i + 1][j - 1] += dp[i][j];
                    dp[i + 1][j - 1] %= MOD;
                } else if j < w && k & (1 << j - 1) > 0 {
                    dp[i + 1][j + 1] += dp[i][j];
                    dp[i + 1][j + 1] %= MOD;
                } else {
                    dp[i + 1][j] += dp[i][j];
                    dp[i + 1][j] %= MOD;
                }
            }
        }
    }
    let ans = dp[h][k];
    println!("{}", ans);
}