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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.util.*;

public class Main {
  public static void main(String[] args) {
    Scanner sc = new Scanner(System.in);
    int h = sc.nextInt();
    int w = sc.nextInt();
    int k = sc.nextInt();
    long MOD = (long)Math.pow(10, 9) + 7;
    long[][] dp = new long[h + 1][w];
    dp[0][0] = 1;
    if(w != 1) {
    for(int i = 0; i < h; i++) {
      for(int j = 0; j < w; j++) {
        for(int t = 0; t < (int)Math.pow(2, w - 1); t++) {
          int[] m = new int[w - 1];
          int p = 0;
          for(int c = 0; c < w - 1; c++) {
            if((t & (1 << c)) != 0) m[c] = 1;
            if((c > 0) && ((m[c - 1] == 1) && (m[c] == 1))) p = 1; 
          }
          if(p == 0) { 
            if(j == 0) {
              if(m[0] == 1) {
                dp[i + 1][j + 1] = (dp[i + 1][j + 1] + dp[i][j]) % MOD;
              } else {
                dp[i + 1][j] = (dp[i + 1][j] + dp[i][j]) % MOD;
              }
            } else if(j == (w - 1)) {
              if(m[w - 2] == 1) {
                dp[i + 1][j - 1] = (dp[i + 1][j - 1] + dp[i][j]) % MOD;
              } else {
                dp[i + 1][j] = (dp[i + 1][j] + dp[i][j]) % MOD;
              }              
            } else {
              if(m[j - 1] == 1) {
                dp[i + 1][j - 1] = (dp[i + 1][j - 1] + dp[i][j]) % MOD;
              } else if(m[j] == 1) {
                dp[i + 1][j + 1] = (dp[i + 1][j + 1] + dp[i][j]) % MOD;
              } else {
                dp[i + 1][j] = (dp[i + 1][j] + dp[i][j]) % MOD;
              }              
            }
          }
        }
      }
    }
    System.out.println(dp[h][k - 1]);
    } else {
      System.out.println(1);
    }
  }
}