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 {
    static int mod = 1000_000_007;
    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[][] dp = new int[h + 1][w + 1];
        dp[0][1] = 1;
        for (int i = 0; i < h; i++) {
            for (int j = 1; j <= w; j++) {
                if (j != 1) dp[i + 1][j - 1] = (int) ((dp[i + 1][j - 1] + (long) dp[i][j] * func(j - 3) * func(w - j - 1)) % mod);
                dp[i + 1][j] = (int) ((dp[i + 1][j] + (long) dp[i][j] * func(j - 2) * func(w - j - 1)) % mod);
                if (j != w) dp[i + 1][j + 1] = (int) ((dp[i + 1][j + 1] + (long) dp[i][j] * func(j - 2) * func(w - j - 2)) % mod);
            }
        }
        System.out.println(dp[h][k]);
    }
    public static int func(int x) {
        if (x <= 0) return 1;
        int[][] dp = new int[x + 1][2];
        dp[0][0] = 1;
        for (int i = 0; i < x; i++) {
            dp[i + 1][0] = dp[i][0] + dp[i][1];
            dp[i + 1][1] = dp[i][0];
        }
        return dp[x][0] + dp[x][1];
    }
}