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) throws Exception {
		Scanner sc = new Scanner(System.in);
		int h = sc.nextInt();
		int w = sc.nextInt();
		int k = sc.nextInt();
		sc.close();

		int mod = 1000000007;
		int[] line = new int[w + 1];
		int[] line0 = new int[w + 2];
		int end = (int) Math.pow(2, w - 1);
		for (int i = 0; i < end; i++) {
			String str = Integer.toBinaryString(i);
			StringBuilder sb = new StringBuilder(str);
			while (sb.length() < w - 1) {
				sb.insert(0, '0');
			}
			if (sb.indexOf("11") == -1) {
				for (int j = 0; j < sb.length(); j++) {
					line[j + 1] += sb.charAt(j) - '0';
					if (j == 0) {
						if (sb.charAt(j) == '0') {
							line0[j + 1]++;
						}
					}
					if (j == sb.length() - 1) {
						if (sb.charAt(j) == '0') {
							line0[j + 2]++;
						}
					} else {
						if (sb.charAt(j) == '0' && sb.charAt(j + 1) == '0') {
							line0[j + 2]++;
						}
					}
				}
			}
		}
//		for (int i = 0; i < line.length; i++) {
//			System.out.print(line[i] + " ");
//		}
//		System.out.println();
//		for (int i = 0; i < line0.length; i++) {
//			System.out.print(line0[i] + " ");
//		}
//		System.out.println();

		long[][] dp = new long[h + 1][w + 2];
		dp[0][1] = 1;
		for (int i = 0; i < h; i++) {
			for (int j = 1; j <= w; j++) {
				// →
				long sum = dp[i][j - 1] * line[j - 1] % mod;
				// ↓
				sum += dp[i][j] * line0[j] % mod;
				sum %= mod;
				// ←
				sum += dp[i][j + 1] * line[j] % mod;
				sum %= mod;
				dp[i + 1][j] = sum;
			}
		}
//		for (int i = 0; i <= h; i++) {
//			for (int j = 1; j <= w; j++) {
//				System.out.print(dp[i][j] + " ");
//			}
//			System.out.println();
//		}
		System.out.println(dp[h][k]);
	}
}