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.IO;
using System.Linq;
using System.Text;
using System.Text.RegularExpressions;

namespace AtCoder
{
	public class ABC
	{
		static int mod = 1000000007;  // 10^9+7

		static void Main(string[] args)
		{
			Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });


			//int n = int.Parse(Console.ReadLine());
			//string s = Console.ReadLine();
			int[] vs = Console.ReadLine().Split().Select(int.Parse).ToArray();
			//long[] vs = Console.ReadLine().Split().Select(long.Parse).ToArray();
			int h = vs[0];
			int w = vs[1];
			int k = vs[2];

			int[,] c = new int[w, 3];
			for (int i = 0; i < 1 << w - 1; i++) {
				bool f = true;
				int t = i;
				for (int j = 0; j < w - 2; j++) {
					if (t % 4 == 3) {
						f = false;
						break;
					}
					t >>= 1;
				}
				if (f) {
					//System.Diagnostics.Debug.WriteLine($"{Convert.ToString(i, 2).PadLeft(w - 1, '0')}");
					t = i << 1;
					for (int j = 0; j < w; j++) {
						int t2 = t % 4;
						if (t2 == 1) {
							c[j, 0]++;  // c[x, 0] x-1 -> x
						} else if (t2 == 0) {
							c[j, 1]++;  // c[x, 1] x   -> x
						} else if (t2 == 2) {
							c[j, 2]++;  // c[x, 2] x+1 -> x
						}
						t >>= 1;
					}
				}
			}

			long[,] dp = new long[w, h + 1];
			dp[k - 1, 0] = 1;
			for (int i = 1; i <= h; i++) {
				for (int j = 0; j < w; j++) {
					if (j > 0) {
						dp[j, i] += dp[j - 1, i - 1] * c[j, 0] % mod;
					}
					dp[j, i] += dp[j, i - 1] * c[j, 1] % mod;
					if (j < w - 1) {
						dp[j, i] += dp[j + 1, i - 1] * c[j, 2] % mod;
					}
					dp[j, i] %= mod;
				}
			}
			//for (int i = 0; i <= h; i++) {
			//	for (int j = 0; j < w; j++) {
			//		System.Diagnostics.Debug.Write($"{dp[j, i]} ");
			//	}
			//	System.Diagnostics.Debug.WriteLine($"");
			//}
			Console.WriteLine(dp[0, h]);


			Console.Out.Flush();
		}
	}
}