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: Wrong Answer)

#include <stdio.h>
int max(int a, int b){return a>b?a:b;}
int main(void){
    int h,w,k,n[8]={1,2,3,6,8,13};
    scanf("%d%d%d",&h,&w,&k);
    if (w==1){
        printf("1\n");
        return 0;
    }
    long long dp[h+1][w], mod=1000000007;
    for (int i=0; i<h+1; i++){
        for (int j=0; j<w; j++){
            dp[i][j]=0;
        }
    }
    dp[0][0]=1;
    for (int i=1; i<h+1; i++){
        for (int j=0; j<w; j++){
            if (j==0){
                dp[i][j]+=(dp[i-1][j]*n[w-2])%mod;
                dp[i][j]+=(dp[i-1][j+1]*n[max(0,w-3)])%mod;
            } else if (j==w-1){
                dp[i][j]+=(dp[i-1][j]*n[w-2])%mod;
                dp[i][j]+=(dp[i-1][j-1]*n[max(0,w-3)])%mod;
            } else {
                dp[i][j]+=(dp[i-1][j]*n[max(0,j-1)]*n[max(0,w-j-2)])%mod;
                dp[i][j]+=(dp[i-1][j-1]*n[max(0,j-2)]*n[max(0,w-j-2)])%mod;
                dp[i][j]+=(dp[i-1][j+1]*n[max(0,j-1)]*n[max(0,w-j-3)])%mod;
            }
        }
    }
    printf("%lld\n",dp[h][k-1]%mod);
	return 0 ;
}