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)


fn func(input_str: &str) {
    let mut iter = input_str.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().unwrap();
    if w == 1 {
        println!("{}", 1);
        return;
    }
    fn rec(a_s: Vec<usize>, pat_0: &mut Vec<usize>, pat_1: &mut Vec<usize>, w: usize) {
        if a_s.len() == w - 1 {
            for i in 1..(w - 1) {
                if a_s[i - 1] + a_s[i] == 0 {
                    pat_0[i] += 1;
                }
            }
            if a_s[0] == 0 {
                pat_0[0] += 1
            }
            if a_s[w - 2] == 0 {
                pat_0[w - 1] += 1
            }
            for i in 0..(w - 1) {
                if a_s[i] == 1 {
                    pat_1[i] += 1;
                }
            }
        } else {
            if let Some(&last) = a_s.last() {
                if last == 0 {
                    let mut n_a_s = a_s.clone();
                    n_a_s.push(1);
                    rec(n_a_s, pat_0, pat_1, w);
                }
            }
            let mut n_a_s = a_s.clone();
            n_a_s.push(0);
            rec(n_a_s, pat_0, pat_1, w);
        }
    }
    let mut pat_0 = vec![0; w];
    let mut pat_1 = vec![0; w - 1];
    rec(vec![0], &mut pat_0, &mut pat_1, w);
    rec(vec![1], &mut pat_0, &mut pat_1, w);
    let mut cnt_s = vec![vec![0; w]; h + 1];
    cnt_s[0][0] = 1;
    for i in 1..(h + 1) {
        for j in 0..w {
            cnt_s[i][j] += cnt_s[i - 1][j] * pat_0[j];
            cnt_s[i][j] %= 1000000007;
            if j < w - 1 {
                cnt_s[i][j] += cnt_s[i - 1][j + 1] * pat_1[j];
                cnt_s[i][j] %= 1000000007;
            }
            if j > 0 {
                cnt_s[i][j] += cnt_s[i - 1][j - 1] * pat_1[j - 1];
                cnt_s[i][j] %= 1000000007;
            }
        }
    }
    println!("{}", cnt_s[h][k-1]);
}

use std::io::Read;

fn main() {
    let mut buf = String::new();
    std::io::stdin().read_to_string(&mut buf).unwrap();
    func(&buf[..]);
}