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::*;
use std::str::FromStr;

#[allow(dead_code)]
fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

const MOD: i64 = 1e+9 as i64 + 7;

fn calc(
    y: usize,
    x: usize,
    pos: usize,
    used: usize,
    memo: &mut Vec<Vec<Vec<Vec<i64>>>>,
    k: usize,
) -> i64 {
    let h = memo.len();
    let w = memo[0].len();
    if y == h {
        return if pos == k { 1 } else { 0 };
    }
    if x == w - 1 {
        return calc(y + 1, 0, pos, 0, memo, k);
    }
    if memo[y][x][pos][used] != -1 {
        return memo[y][x][pos][used];
    }
    let mut ret = 0;
    if used == 0 {
        let npos = match pos {
            _ if pos == x => pos + 1,
            _ if pos == x + 1 => pos - 1,
            _ => pos,
        };
        ret = (ret + calc(y, x + 1, npos, 1, memo, k)) % MOD;
    }
    ret = (ret + calc(y, x + 1, pos, 0, memo, k)) % MOD;
    memo[y][x][pos][used] = ret;
    return ret;
}

fn main() {
    let h = read::<usize>();
    let w = read::<usize>();
    let k = read::<usize>() - 1;
    let mut memo = vec![vec![vec![vec![-1; 2]; w]; w]; h];
    let ans = calc(0, 0, 0, 0, &mut memo, k);
    println!("{}", ans);
}