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)

macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

//
// # Examples
//
// # integers
// input! { n: usize, m: usize, ... }
//
// # array
// input! { n: usize, v: [i32; n], }
//
// # matrix
// input! { row: usize, column: usize, mat: [[i32; column]; row] }
//
// # string
// input! { s: chars }
// 
// # maze
// input!{
//     h: usize,
//     w: usize,
//     board: [chars; h],
// }
// 3 4
// ####
// #..#
// .#.#

// https://practice.contest.atcoder.jp/tasks/practice_1
//fn main() {
//    input!{
//        a: usize, b: usize, c: usize,
//        foo: chars,
//    }
//    let s: String = foo.into_iter().collect();
//    println!("{} {}", a+b+c, s);
//}

fn fib(n: usize) -> usize {
    if n == 0 { return 1; }
    match n {
        1|2 => n,
        _ => fib(n-1) + fib(n-2)
    }
}

fn main() {
    input! { h: usize, w: usize, k: usize }

    // dp[h][k]: the number that result in k-th line if amida have been traversed by h-th row
    let mut dp = [[0; 8+1]; 100+1];

    for i in 1..h+1 {
        for j in 1..w+1 {
            if i == 1 {
                if j == 1 || j == 2 { dp[i][j] = fib(w-j); }
            } else {
                // to straight
                if w >= j {
                    dp[i][j] += dp[i-1][j] * fib(j-1) * fib(w-j) % 1_000_000_007;
                }
                // to left
                if w-j >= 1 {
                    dp[i][j] += dp[i-1][j+1] * fib(j-1) * fib(w-j-1) % 1_000_000_007;
                }
                // to right
                if j >= 2 {
                    dp[i][j] += dp[i-1][j-1] * fib(j-2) * fib(w-j) % 1_000_000_007;
                }
            }
        }
    }
    println!("{}", dp[h][k] % 1_000_000_007);
}