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.*;
class Main
{
    static int mod = 1000000007;
    static int h, w, k;
    static int[] p;
    static long[][] dp;
    public static void main (String[] args)
    {
        Scanner sc = new Scanner(System.in);
        h = sc.nextInt();
        w = sc.nextInt();
        k = sc.nextInt();
        p = new int[w + 2];
        p[0] = 1; p[1] = 1;
        for(int i = 2; i < w + 2; i++){
            p[i] = p[i-1] + p[i-2];
        }
        dp = new long[h+1][w+1];
        for(int i = 1; i <= h; i++){
            for(int j = 1; j <= w; j++){
                dp[i][j] = calc(i, j);
            }
        }
        System.out.println(dp[h][k]);
    }
    public static long calc(int height, int width){
        if(height == 1 && width == 1) return p[w - 1];
        else if(height == 1 && width == 2) return p[w-2];
        else if(height + 1< width) return 0;
        else{
            long res = 0;
            res += dp[height - 1][width] * p[width - 1] * p[w - width];
            res %= mod;
            if(width > 1){
                res += dp[height - 1][width - 1] * p[width - 2] * p[w - width];
                res %= mod;
            }
            if(width < w){
                res += dp[height - 1][width + 1] * p[width - 1] * p[w - 1 - width];
                res %= mod;
            }
            return res;
        }
    }
}