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 reg register
using namespace std;
typedef long long ll;
const int mod=1e9+7;
int h,w,k;
int f[105][10],fib[205];
int main(){
	cin>>h>>w>>k;fib[1]=fib[2]=1;
	for(reg int i=3;i<=200;i++)fib[i]=(fib[i-1]+fib[i-2])%mod;
	f[h+1][1]=1;
	for(reg int i=h+1;i>1;i--){
		for(reg int j=1;j<=w;j++){
			if(j>1)f[i-1][j-1]=(f[i-1][j-1]+1ll*f[i][j]*fib[j-1]%mod*fib[w-j+1]%mod)%mod;
			f[i-1][j]=(f[i-1][j]+1ll*f[i][j]*fib[j]%mod*fib[w-j+1]%mod)%mod;
			if(j<w)f[i-1][j+1]=(f[i-1][j+1]+1ll*f[i][j]*fib[j]%mod*fib[w-j]%mod)%mod;
		}
	}printf("%d\n",f[1][k]);
	return 0;
}