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: Runtime Error)


import java.util.HashSet;
import java.util.Scanner;
import java.util.Set;

public class Main {

	public static Set<boolean[]> amida = new HashSet<>();
	public static boolean[] amicre;
	public static int H, W, K;

	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		H = sc.nextInt();
		W = sc.nextInt();
		K = sc.nextInt();
		create(0);
		amida.removeIf(o -> filter(o));
		long count = 0;
		for(boolean[] ami : amida) {
			if(K == play(ami)) {
				count++;
			}
		}
		System.out.println(count % 1000000007);
	}

	public static int play(boolean[] ami) {
		int tmp = 1;
		while(tmp <= H * W) {
			if(ami[tmp - 1]) {
				tmp--;
			} else if(ami[tmp]) {
				tmp++;
			}
			tmp += W;
		}
		int ret = 0;
		if(tmp % W == 0) {
			ret = W;
		} else {
			ret = tmp % W;
		}
		return ret;
	}

	public static boolean filter(boolean[] ami) {
		boolean ans = false;
		for(int i = 0; i < H * W; i += W) {
			if(ami[i]) {
				ans = true;
				break;
			}
		}
		if(!ans) {
			for(int i = 1; i < H * W; i++) {
				if(ami[i] && (ami[i - 1] || ami[i + 1])) {
					ans = true;
					break;
				}
			}
		}
		return ans;
	}

	public static void create(int i) {
		if(i <= H * W) {
			amicre[i] = true;
			create(i + 1);
			amicre[i] = false;
			create(i + 1);
		} else {
			amida.add(amicre);
		}
	}

}