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 <bits/stdc++.h>
#define r(i,n) for(int i=0;i<n;i++)
#define int long long
using namespace std;
typedef pair<int,int>P;
typedef pair<int,P>P2;
typedef pair<int,string>XXX;
#define F first
#define S second

int dp[101][9];

signed main(){
    int h,w,k;
    cin>>h>>w>>k;
    dp[0][0]=1;
    for(int i=0;i<h;i++){
        for(int j=0;j<(1<<w-1);j++){
            int f=0;
            for(int x=0;x<w;x++){
                if((j&(1<<x))&&(j&(1<<(x+1))))f++;
            }
            //cout<<j<<' '<<f<<endl;
            if(f)continue;
            for(int x=0;x<w;x++){
                if(x&&(j&(1<<(x-1))))dp[i+1][x-1]+=dp[i][x],dp[i+1][x-1]%=1000000007;
                else if(j&(1<<(x)))dp[i+1][x+1]+=dp[i][x],dp[i+1][x+1]%=1000000007;
                else dp[i+1][x]+=dp[i][x],dp[i+1][x]%=1000000007;
            }
        }
        //cout<<dp[i+1][0]<<' '<<dp[i+1][1]<<' '<<dp[i+1][2]<<endl;
    }
    cout<<dp[h][k-1]<<endl;
}