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;
class Program
{
	static void Main(string[] args)
	{
		string[] str = Console.ReadLine().Split();
		long H = long.Parse(str[0]);
		long W = long.Parse(str[1]);
		long K = long.Parse(str[2]);
		long[] C = new long[7]{2,3,5,8,13,21,34};
		long mod = 1000000007;
		long[,]A = new long[H+1,W];
		A[0,0] = 1;
		for(var i=1;i<H+1;i++){
			for(var j=0;j<W;j++){
				long a = 1;
				if(j>0){
					if(j>=3){
						a *= C[j-3];
					}
					if(W-j>=3){
						a *= C[W-j-3];
					}
					A[i,j-1] += A[i-1,j] * a;
				}
				a = 1;
				if(j>=2){
					a *= C[j-2];
				}
				if(W-j>=3){
					a *= C[W-j-3];
				}
				A[i,j] += A[i-1,j] * a;
				a = 1;
				if(j<W-1){
					if(j>=2){
						a *= C[j-2];
					}
					if(W-j>=4){
						a *= C[W-j-4];
					}
					A[i,j+1] += A[i-1,j] * a;
				}
			}
			for(var j=0;j<W;j++){
				A[i,j] %= mod;
			}
		}
		Console.WriteLine(A[H,K-1]%mod);
	}
}