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::*;
use std::collections::HashMap;

const MOD:usize = 1_000_000_007;
 
#[fastout]
fn main() {
  input! {
    h: usize,
    w: usize,
    kk: usize
  }
  
  let mut dp: Vec<Vec<usize>> = vec![vec![0;w];h+1];
  dp[0][0] = 1;
  
  for r in 0..h {
    for c in 0..w {
      for k in 0..(1 << (w-1)) {
        let mut ok = true;
        
        let w = if w < 2 {
          2
        } else {
          w
        };
        
        for l in 0..(w-2) {
          if (k >> l) & 1 == 1 && k >> l+1 & 1 == 1 {
            ok = false;
          }
        }
        
        if !ok { continue }
        if c >= 1 && k >> (c - 1) & 1 == 1 {
          dp[r+1][c-1] += dp[r][c];
          dp[r+1][c-1] %= MOD;
        }
        else if c <= w-2 && k >> c & 1 == 1 {
          dp[r+1][c+1] += dp[r][c];
          dp[r+1][c+1] %= MOD;
        }
        else {
          dp[r+1][c] += dp[r][c];
          dp[r+1][c] %= MOD;
        } 
      }
    }
  }
  
  println!("{}", dp[h][kk-1]);
}