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::cmp;
use std::collections::BinaryHeap;

fn read_line() -> String {
    let mut buf = String::new();
    io::stdin().read_line(&mut buf).ok();
    buf
}

const MOD: u64 = 1_000_000_007;

fn main() {
    let line: Vec<usize> = read_line().split_whitespace().map(|s| s.parse().unwrap()).collect();
    let h = line[0];
    let w = line[1];
    let k = line[2];

    let mut p = vec![vec![0; w+2]; w+2];
    let mut q = vec![0u64; w+2];
    q[0] = 1;
    q[1] = 1;
    for i in 0..w {
        q[i+2] = q[i+1] + q[i] % MOD;
    }

    for i in 1..w+1 {
        for j in 1..w+1 {
            if i+1 == j {
                p[i][j] = q[i-1] * q[w-j] % MOD;
            } else if i==j+1 {
                p[i][j] = q[w-i] * q[j-1] % MOD;
            } else if i==j {
                p[i][i] = q[w-i] * q[i-1] % MOD;
            }
        }
    }

    let mut memo = vec![vec![0; w]; h + 2];
    memo[0][0] = 1;

    for i in 0..h {
        for t in 0..w {
            for j in 0..w {
                memo[i+1][t] += memo[i][j] * p[j+1][t+1] % MOD;
                memo[i+1][t] %= MOD;
            } 
        }
    }
    println!("{}", memo[h][k-1]);
}