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 final int MOD = 1000000007;
	public static void main(String[] args){
		Scanner sc = new Scanner(System.in);
		int h = sc.nextInt();
		int w = sc.nextInt();
		int kk = sc.nextInt();
		long[][] dp = new long[h + 1][w];
		dp[0][0] = 1;
		for (int i = 1; i <= h; i++) {
	        for (int k = 0; k < Math.pow(2, w - 1); k++) {
	            boolean[] bars = toBitArray(k, w - 1);
	            if (bars == null) {
	                continue;
	            }
    		    for (int j = 0; j < w; j++) {
		            if (j != 0 && bars[j - 1]) {
		                dp[i][j] += dp[i - 1][j - 1];
		            } else if (j != w - 1 && bars[j]) {
		                dp[i][j] += dp[i - 1][j + 1];
		            } else {
		                dp[i][j] += dp[i - 1][j];
		            }
		            dp[i][j] %= MOD;
		        }
		    }
		}
		System.out.println(dp[h][kk - 1]);
	}
	
	static boolean[] toBitArray(int x, int size) {
	    boolean[] ans = new boolean[size];
	    boolean prev = false;
	    int idx = 0;
	    while (x > 0) {
	        ans[idx] = (x % 2 == 1);
	        if (ans[idx] && prev) {
	            return null;
	        }
	        prev = ans[idx];
	        x /= 2;
	        idx++;
	    }
	    return ans;
	}
	
}