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.Scanner;

public class Main {
	
	public static void main(String[] args) {
		long inf = 1000000007;
		Scanner in = new Scanner(System.in);
		int h = in.nextInt();
		int w = in.nextInt();
		int k = in.nextInt();
		if(w==1) {
			System.out.println(1);
		}else {
			int[] fib = new int[w];
			fib[0] = 1;
			fib[1] = 2;
			for(int i=2;i<w;i++) {
				fib[i] = fib[i-1] + fib[i-2];
			}
			int[][] puttern = new int[w][3];
			puttern[0][0] = 0;
			puttern[0][1] = fib[w-2];
			puttern[0][2] = w-3>=0?fib[w-3]:1;
			for(int i=1;i<w-1;i++) {
				puttern[i][0] = (i-2>=0?fib[i-2]:1) * (w-i-2>=0?fib[w-i-2]:1);
				puttern[i][1] = (i-1>=0?fib[i-1]:1) * (w-i-2>=0?fib[w-i-2]:1);
				puttern[i][2] = (i-1>=0?fib[i-1]:1) * (w-i-3>=0?fib[w-i-3]:1);
			}
			puttern[w-1][0] = w-3>=0?fib[w-3]:0;
			puttern[w-1][1] = fib[w-2];
			puttern[w-1][2] = 0;
			
			long[][] dp = new long[h+1][w];
			dp[0][0] = 1;
			for(int i=0;i<h;i++) {
				dp[i+1][0] = (dp[i][0]*puttern[0][1]%inf + dp[i][1]*puttern[0][2]%inf)%inf;
				for(int j=1;j<w-1;j++) {
					dp[i+1][j] = (dp[i][j-1]*puttern[j][0]%inf + dp[i][j]*puttern[j][1]%inf + dp[i][j+1]*puttern[j][2]%inf)%inf;
				}
				dp[i+1][w-1] = (dp[i][w-2]*puttern[w-1][0]%inf + dp[i][w-1]*puttern[w-1][1]%inf)%inf;
			}
			System.out.println(dp[h][k-1]);
		}
		
		
		
	}

}