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.Linq;

namespace ConsoleApp1
{
	class Program
	{
		static void Main(string[] a_asArgs)
		{
			byte[][] l_aaiPattern = new byte[][] {
				new byte[] {0x00, 0x01},
				new byte[] {0x00, 0x01, 0x02},
				new byte[] {0x00, 0x01, 0x02, 0x04, 0x05},
				new byte[] {0x00, 0x01, 0x02, 0x04, 0x05, 0x08, 0x09, 0x0a},
				new byte[] {0x00, 0x01, 0x02, 0x04, 0x05, 0x08, 0x09, 0x0a, 0x10, 0x11, 0x12, 0x14, 0x15},
				new byte[] {0x00, 0x01, 0x02, 0x04, 0x05, 0x08, 0x09, 0x0a, 0x10, 0x11, 0x12, 0x14, 0x15, 0x20, 0x21, 0x22, 0x24, 0x25, 0x28, 0x29, 0x2a},
				new byte[] {0x00, 0x01, 0x02, 0x04, 0x05, 0x08, 0x09, 0x0a, 0x10, 0x11, 0x12, 0x14, 0x15, 0x20, 0x21, 0x22, 0x24, 0x25, 0x28, 0x29, 0x2a, 0x40, 0x41, 0x42, 0x44, 0x45, 0x48, 0x49, 0x4a, 0x50, 0x51, 0x52, 0x54, 0x55}
			};

			int[] l_aiInput = Console.ReadLine().Split(' ').Select(l_sItem => int.Parse(l_sItem)).ToArray();
			int l_iH = l_aiInput[0];
			int l_iW = l_aiInput[1];
			int l_iK = l_aiInput[2];

			if (l_iW == 1)
			{
				Console.WriteLine("1");
				return;
			}

			byte[] l_aiPattern = l_aaiPattern[l_iW - 2];

			uint[] l_aiUpper = new uint[l_iW];
			uint[] l_aiLower = new uint[l_iW];
			l_aiUpper[0] = 1;

			for (int l_iY = 0; l_iY < l_iH; l_iY++)
			{
				foreach (byte l_iPattern in l_aiPattern)
				{
					for (int l_iX = 0; l_iX < l_iW; l_iX++)
					{
						if (l_iX > 0 && (l_iPattern & (1 << l_iX - 1)) != 0)
						{
							l_aiLower[l_iX] += l_aiUpper[l_iX - 1];
						}
						else if (l_iX < l_iW - 1 && (l_iPattern & (1 << l_iX)) != 0)
						{
							l_aiLower[l_iX] += l_aiUpper[l_iX + 1];
						}
						else
						{
							l_aiLower[l_iX] += l_aiUpper[l_iX];
						}
						l_aiLower[l_iX] %= 1000000007;
					}
				}
				l_aiLower.CopyTo(l_aiUpper, 0);
				Array.Clear(l_aiLower, 0, l_iW);
			}
			Console.WriteLine(l_aiUpper[l_iK - 1]);
		}
	}
}