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)

#include<stdio.h>
int main() {
	long long int h,w,k,memo,result;
	long long int two[10] = { 1,2,3,5,8,13,21,34,55,256,512 };
	scanf("%lld%lld%lld", &h, &w, &k);
	long long int s[105][10] = {};
	s[h + 1][k] = 1;
	for (int i = h; i >= 1; i--) {
		for (int j = 1; j <= w; j++) {
			memo = 0;
			if (j == 1 || j == w) {
				memo += two[w - 2] * s[i + 1][j];
			}
			else {
				memo += (two[j-2]*two[w-j-1]) * s[i + 1][j];
			}

			if (j == 1||j==w-1) {
				memo += two[w - 3] * s[i + 1][j+1];
			}
			else if (j == w) {

			}
			else
			{
				memo += (two[j-2]*two[w - j-2]) * s[i + 1][j+1];
			}

			if (j == 1) {
				
			}
			else if (j == w||j==2) {
				memo += two[w - 3] * s[i + 1][j - 1];
			}
			else
			{
				memo += (two[j - 3] * two[w - j - 1]) * s[i + 1][j-1];
			}
			memo %= 1000000007;
			s[i][j] = memo;
		}
	}
	printf("%lld", s[1][1]);

}