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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <iostream>
#include <algorithm>
#include <vector>
#include <stack>
#include <queue>
#include <set>
#include <string>
#include <iomanip>
using namespace std;
long long mod=1e9+7;
long long c[8]={1,1,2,3,5,8,13,21};
int main(){
	int h,w,k;
	cin>>h>>w>>k;
	long long dp[101][8]={};
	dp[0][0]=1;
	for(int i=1;i<=h;i++){
		for(int j=0;j<w;j++){
			if(j==0){
				dp[i][0]+=dp[i-1][0]*c[w-1];
				if(w>=2){
					dp[i][0]+=dp[i-1][1]*c[w-2];
				}
			}else if(j==w-1){
				dp[i][w-1]+=dp[i-1][w-1]*c[w-1];
				if(w>=2){
					dp[i][w-1]+=dp[i-1][w-2]*c[w-2];
				}
			}else{
				dp[i][j]+=dp[i-1][j]*c[j]*c[w-j-1];
				dp[i][j]+=dp[i-1][j-1]*c[j-1]*c[w-j-1];
				dp[i][j]+=dp[i-1][j+1]*c[j]*c[w-j-2];
			}
			dp[i][j]%=mod;
		}
	}
	cout<<dp[h][k-1]<<endl;
	return 0;
}