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)

#[allow(unused_imports)]
use template::buf_scan::*;
#[allow(unused_imports)]
use template::*;

#[allow(dead_code)]
#[macro_use]
pub mod macros {
    #[macro_export]
    macro_rules! debug {
        ($($a:expr),*) => {{
            use ::std::io::Write;
            writeln!(&mut ::std::io::stderr(), concat!($(stringify!($a), " = {:?}, "),*), $($a),*).unwrap();
        }}
    }

    #[macro_export]
    macro_rules! input {
        (source = $s:expr, $($r:tt)*) => {
            let mut scanner = $crate::template::buf_scan::BufScanner::new($s.as_bytes());
            input_inner!{scanner, $($r)*}
        };
        ($($r:tt)*) => {
            let stdin = ::std::io::stdin();
            let mut scanner = $crate::template::buf_scan::BufScanner::new(stdin.lock());
            input_inner!{scanner, $($r)*}
        };
    }

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

    macro_rules! read_value {
        ($scanner:ident, ( $($t:tt),* )) => {
            ( $(read_value!($scanner, $t)),* )
        };
        ($scanner:ident, [ $t:tt ; $len:expr ]) => {
            (0..$len).map(|_| read_value!($scanner, $t)).collect::<Vec<_>>()
        };
        ($scanner:ident, chars) => {
            read_value!($scanner, String).chars().collect::<Vec<char>>()
        };
        ($scanner:ident, usize1) => {
            read_value!($scanner, usize) - 1usize
        };
        ($scanner:ident, $t:ty) => {
            ($scanner).next::<$t>()
        };
    }
}

fn main() {
    input! {
        h: usize,
        w: usize,
        k: usize1,
    }
    let modulo: u64 = 1_000_000_007u64;

    let valid_horizontal: Vec<_> = (0..(1<<(w-1))).filter(|a| {
        (0..w).all(|i|{ 3 != ((a>>i)&3)})
    }).collect();
    let mut dp: Vec<u64> = vec![0u64; w];
    dp[0] = 1;
    for _ in 0..h {
        let mut qb: Vec<u64> = vec![0u64; w];
        for A in valid_horizontal.as_slice() {
            for i in 0..w {
                if i < w-1 && 1 == (A>>i&1) {
                    qb[i+1] += dp[i];
                    qb[i+1] %= modulo;
                } else if i > 0 && 1 == (A>>(i-1)&1) {
                    qb[i-1] += dp[i];
                    qb[i-1] %= modulo;
                } else {
                    qb[i] += dp[i];
                    qb[i] %= modulo;
                }
            }
        }
        dp = qb;
    }
    println!("{}", dp[k]);
}

#[allow(dead_code)]
pub mod template {
    pub mod buf_scan {
        pub struct BufScanner<R: ::std::io::BufRead> {
            br: R,
            buf: Vec<u8>,
            pos: usize,
        }

        impl<R: ::std::io::BufRead> BufScanner<R> {
            pub fn new(br: R) -> Self {
                BufScanner {
                    br: br,
                    buf: Vec::new(),
                    pos: 0usize,
                }
            }
            fn fill_line(&mut self) {
                self.buf.clear();
                self.pos = 0;
                self.br
                    .read_until(b'\n', &mut self.buf)
                    .expect("Read a line");
                if let Some(&b'\n') = self.buf.last() {
                    self.buf.pop();
                    if let Some(&b'\r') = self.buf.last() {
                        self.buf.pop();
                    }
                }
            }
            fn skip_space(&mut self) {
                loop {
                    for i in self.pos..self.buf.len() {
                        if self.buf[i] != b' ' {
                            self.pos = i;
                            return;
                        }
                    }
                    self.fill_line();
                }
            }
            pub fn read_token(&mut self) -> &str {
                self.skip_space();
                let start = self.pos;
                while self.pos < self.buf.len() && self.buf[self.pos] != b' ' {
                    self.pos += 1;
                }
                let ret = unsafe { ::std::str::from_utf8_unchecked(&self.buf[start..self.pos]) };
                self.pos += 1; // skip the last space (or len() + 1 which is fine)
                return ret;
            }
            pub fn next<T: ::std::str::FromStr>(&mut self) -> T {
                self.read_token().parse::<T>().ok().expect("Parse")
            }
        }
    }
}