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: Wrong Answer)

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();
        int mod = 1000000007;

        if(w == 1){
            System.out.println("1");
            System.exit(0);
        }

        long[][] dp = new long[h+1][w];

        dp[0][0] = 1;
        for(int i = 0;i < w;i++){
            dp[0][w-1] = 0;
        }

        int[] p = new int[w-1];
        int sum = 0;
        for(int i =0;i < Math.pow(2, w-1);i++){
            if((i & (i << 1)) != 0){
                continue;
            }
            sum++;
            for(int j = 0;j < w;j++){
                if((i & (1 << j)) != 0){
                    p[j]++;
                }
            }
        }

        for(int i = 1;i <= h;i++){
            dp[i][0] = dp[i-1][0] * (sum-p[0]) + dp[i-1][1] * p[0]%mod;
            dp[i][w-1] = dp[i-1][w-1] * (sum-p[w-2]) + dp[i-1][w-2] * p[w-2]%mod;
            for(int j = 1;j < w-1;j++){
                dp[i][j] = (dp[i-1][j-1]*p[j-1] + dp[i-1][j] * (sum-p[j-1]-p[j]) + dp[i-1][j+1]*p[j])%mod;
            }
        }
        System.out.println(dp[h][k-1]);
    }

}