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)

use proconio::input;

#[proconio::fastout]
fn main() {
    input! {
        H: i128,
        W: usize,
        K: usize
    }
    if W == 1 {
        println!("1");
        return;
    }
    const MOD: i128 = 1_000_000_007;

    let mut positions = vec![0i128; W];
    positions[0] = 1;
    for _y in 0..H {
        let mut row = vec![0i128; W];
        for x in 0usize..(1 << (W - 1)) {
            // 同じ端点を共有してはならない
            if x & (x << 1) != 0 { continue; }

            // iビット目が立っているとき、i番目の橋が存在する
            // ここでi番目の橋とは、i番目の線から右に伸びるもの

            for i in 0..W {
                // i番目の橋があるなら
                if x & (1 << i) != 0 {
                    // 左から右へ
                    row[i + 1] += positions[i];
                    row[i + 1] %= MOD;
                    // 右から左へ
                    row[i] += positions[i + 1];
                    row[i] %= MOD;
                } else if i == 0 || x & (1 << (i - 1)) == 0 {
                    // そのまま下へ
                    row[i] += positions[i];
                    row[i] %= MOD;
                }
            }
        }
        positions = row;
    }
    println!("{}", positions[K - 1]);
}