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.*;
import java.awt.*;

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 e = 1000000007;

        long[][] dp = new long[h+1][w];
        dp[0][0] = 1;

        for (int i = 1; i <= h; i++) {
            for (int j = 0; j < w; j++) {
                dp[i][j] += (dp[i-1][j] * tmp(w-j-2) * tmp(j-1)) % e;
                if (j-1 >= 0) {
                    dp[i][j] += (dp[i-1][j-1] * tmp(w-j-2) * tmp(j-2)) % e;
                }
                if (j+1 < w) {
                    dp[i][j] += (dp[i-1][j+1] * tmp(w-j-3) * tmp(j-1)) % e;
                }
                dp[i][j] %= e;
            }
        }

        System.out.println(dp[h][k-1]);
    }

    public static long tmp(int x) {
        long ret = 1;

        if (x == 1) {
            ret = 2;
        } else if (x == 2) {
            ret = 3;
        } else if (x == 3) {
            ret = 5;
        } else if (x == 4) {
            ret = 8;
        } else if (x == 5) {
            ret = 13;
        } else if (x == 6) {
            ret = 21;
        }

        return ret;
    }
}