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 C (GCC 5.4.1) (judge verdict: Accepted)

/* Coached by rainboy */
#include <stdio.h>

#define H 100
#define W 8
#define MD 1000000007

int main() {
	static int ff[W + 1], dp[H + 1][W];
	int h, w, k, i, j;

	scanf("%d%d%d", &h, &w, &k), k--;
	ff[0] = ff[1] = 1;
	for (i = 2; i <= w; i++)
		ff[i] = (ff[i - 1] + ff[i - 2]) % MD;
	dp[0][0] = 1;
	for (i = 0; i < h; i++)
		for (j = 0; j < w; j++) {
			long long x = dp[i][j];

			dp[i + 1][j] = (dp[i + 1][j]
					+ x * ff[j] % MD * ff[w - 1 - j]) % MD;
			if (j > 0)
				dp[i + 1][j - 1] = (dp[i + 1][j - 1]
						+ x * ff[j - 1] % MD * ff[w - 1 - j]) % MD;
			if (j + 1 < w)
				dp[i + 1][j + 1] = (dp[i + 1][j + 1]
						+ x * ff[j] % MD * ff[w - 1 - j - 1]) % MD;
		}
	printf("%d\n", dp[h][k]);
	return 0;
}