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 (1.42.0) (judge verdict: Accepted)

#![allow(unused_imports)]

use proconio::{input, fastout};
use proconio::marker::*;

fn main() {
    input! {
    h: usize,
    w: usize,
    t: Usize1
    }
    let m = 1_000_000_007;
    let mut dp = vec![vec![0; w]; h + 1];
    dp[0][0] = 1;
    let p: Vec<usize> = (0..w).collect();
    let mut ps = vec![];
    for i in 0..1 << (w - 1) {
        let mut np = p.clone();
        let mut flag = true;
        for j in 0..(w - 1) {
            if i & 1 << j > 0 && i & 2 << j > 0 {
                flag = false;
                break;
            }
        }
        if flag {
            for j in 0..(w - 1) {
                if i & 1 << j > 0 {
                    np.swap(j, j + 1);
                }
            }
            ps.push(np);
        }
    }
    for i in 0..h {
        for j in 0..ps.len() {
            for k in 0..w {
                dp[i + 1][k] += dp[i][ps[j][k]];
                dp[i + 1][k] %= m;
            }
        }
    }
    let ans = dp[h][t];
    println!("{}", ans);
}