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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
class Program{
	static void Main(){
		const int mod=1000000007;
		int[] s=Array.ConvertAll(Console.ReadLine().Split(),int.Parse);
		long[] h=new long[s[1]-1];
		long[,] ans=new long[s[0]+1,s[1]];
		long z=1;
		for(int i=1;i<(1<<(s[1]-1));i++){
			string st=new string('0',s[1]-1-Convert.ToString(i,2).Length)+Convert.ToString(i,2);
			bool b=true;
			for(int j=1;j<s[1]-1;j++){
				if(st[j-1]=='1'&&st[j]=='1'){
					b=false;
					break;
				}
			}
			if(b){
				z++;
				for(int j=0;j<s[1]-1;j++){
					if(st[j]=='1'){h[j]++;}
				}
			}
		}
		ans[0,0]++;
		if(s[1]>1){
		for(int i=1;i<=s[0];i++){
			for(int j=0;j<s[1];j++){
				if(j==0){
					ans[i,0]+=ans[i-1,0]*(z-h[0]);
					ans[i,0]%=mod;
					ans[i,1]+=ans[i-1,0]*(h[0]);
					ans[i,1]%=mod;
				}
				else if(j==s[1]-1){
					ans[i,j-1]+=ans[i-1,j]*(h[j-1]);
					ans[i,j-1]%=mod;
					ans[i,j]+=ans[i-1,j]*(z-h[j-1]);
					ans[i,j]%=mod;
				}
				else{
					ans[i,j-1]+=ans[i-1,j]*(h[j-1]);
					ans[i,j-1]%=mod;
					ans[i,j]+=ans[i-1,j]*(z-h[j-1]-h[j]);
					ans[i,j]%=mod;
					ans[i,j+1]+=ans[i-1,j]*(h[j]);
					ans[i,j+1]%=mod;
				}
			}

			
		}
		Console.WriteLine(ans[s[0],s[2]-1]);
		}
		else{Console.WriteLine(1);}
	}
}