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{
	public static void main(String args[]){
		int mod = 1000000007;
		Scanner sc = new Scanner(System.in);
		int H = sc.nextInt();
		int W = sc.nextInt();
		int K = sc.nextInt();
		
		if(W==1){
			System.out.print("1");
			return;
		}
		
		long[][] dp = new long[H+1][W+1];
		dp[0][1] = 1;
		for(int i=2; i<=W; i++){
			dp[0][i] = 0;
		}
		for(int i=1; i<=H; i++){
			dp[i][1] = ((dp[i-1][1]*f(W-2)) + (dp[i-1][2]*f(W-3)))%mod;
			dp[i][W] = ((dp[i-1][W]*f(W-2)) + (dp[i-1][W-1]*f(W-3)))%mod;
			for(int j=1; j<=W-1; j++){
				dp[i][j] = ((dp[i-1][j-1]*f(j-3)*f(W-j-1)) + (dp[i-1][j]*f(j-2)*f(W-j-1)) + (dp[i-1][j+1]*f(j-2)*f(W-j-2)))%mod;
			}
		}
		System.out.print(dp[H][K]);
		return;
	}

	public static int f(int a){
		if(a<0){
			return 1;
		}
		switch(a){
			case 0: return 1;
			case 1: return 2;
			case 2: return 3;
			case 3: return 5;
			case 4: return 8;
			case 5: return 13;
			case 6: return 21;
			case 7: return 34;
			default: return 0;
		}
	}
}