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: Wrong Answer)

using System;
using System.Linq;

namespace abc113_d
{
	class Program
	{
		static void Main(string[] args)
		{
			long[] x = Console.ReadLine().Split().Select(long.Parse).ToArray();
			long[,] dp = new long[x[0] + 2, x[1]]; dp[0, 0] = 1;
			long mod = 1000000007;
			for (int i = 0; i <= x[0]; i++)
			{
				for (int j = 0; j < x[1]; j++)
				{
					if (j > 0)
					{
						if (j - 1 == 0 || j == x[1] - 1)
							dp[i + 1, j - 1] += dp[i, j] * Math.Max(1, 1 << (int)x[1] - 3);
						else
							dp[i + 1, j - 1] += dp[i, j] * Math.Max(1, 1 << (int)x[1] - 4);
						dp[i + 1, j - 1] %= mod;
					}
					if (j == 0 || j == x[1] - 1)
						dp[i + 1, j + 0] += dp[i, j] * Math.Max(1, 1 << (int)x[1] - 2);
					else
						dp[i + 1, j + 0] += dp[i, j] * Math.Max(0, 1 << (int)x[1] - 3);
					dp[i + 1, j + 0] %= mod;
					if (j < x[1] - 1)
					{
						if (j == 0 || j + 1 == x[1] - 1)
							dp[i + 1, j + 1] += dp[i, j] * Math.Max(1, 1 << (int)x[1] - 3);
						else
							dp[i + 1, j + 1] += dp[i, j] * Math.Max(1, 1 << (int)x[1] - 4);
						dp[i + 1, j + 1] %= mod;
					}
				}
			}
			Console.WriteLine(dp[x[0], x[2] - 1]);
		}
	}
}