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)

const MOD: usize = 1_000_000_007;

fn memo(cur_h: usize, cur_w: usize, cur_goal: usize, dp: &mut Vec<Vec<Vec<Option<usize>>>>, limit_h: usize, limit_w: usize, target_goal: usize) -> usize {
    if cur_h >= limit_h {
        if cur_goal == target_goal {
            return 1;
        }
        return 0;
    }
    if let Some(memo_value) = dp[cur_h][cur_w][cur_goal] {
        return memo_value;
    }
    let mut result: usize = 0;
    //use
    {
        let mut next_w = cur_w + 2;
        let mut next_h = cur_h;
        let mut next_goal = cur_goal;
        if next_w >= limit_w {
            next_w = 0;
            next_h += 1;
        }
        let left = cur_w;
        let right = cur_w + 1;
        if left == cur_goal {
            next_goal = right;
        } else if right == cur_goal {
            next_goal = left;
        }

        result += memo(next_h, next_w, next_goal, dp, limit_h, limit_w, target_goal);
        result %= MOD;
    }
    //no use
    {
        let mut next_w = cur_w + 1;
        let mut next_h = cur_h;
        if next_w >= limit_w {
            next_w = 0;
            next_h += 1;
        }
        result += memo(next_h, next_w, cur_goal, dp, limit_h, limit_w, target_goal);
        result %= MOD;
    }
    dp[cur_h][cur_w][cur_goal] = Some(result);
    return result;
}

fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let h: usize = sc.read();
    let w: usize = sc.read::<usize>() - 1;
    let k: usize = sc.read::<usize>() - 1;
    //dp[h][w][goal]
    let mut dp: Vec<Vec<Vec<Option<usize>>>> = vec![vec![vec![None; 10]; 10]; 110];
    let mut res = 1;
    if w != 0 {
        res = memo(0, 0, 0, &mut dp, h, w, k);
    }
    println!("{}", res);
}

//snippet from kenkoooo
pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self.stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn read_line(&mut self) -> String {
        use std::io::Read;
        let buf = self.stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b'\n' || b == b'\r')
            .take_while(|&b| b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }

    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}