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;
using System.Text;
using System.Threading.Tasks;

namespace ABC113D
{
    class Program
    {
        static long[,] dp;
        static int[] fibonucchi = new int[9];
        static void Main(string[] args)
        {
            var HWK = Console.ReadLine().Split().Select(int.Parse).ToArray();
            int H = HWK[0];
            int W = HWK[1];
            int K = HWK[2];
            dp = new long[W, H + 1];
            dp[0, 0] = 1;
            long div = 1000000007;

            fibonucchi[0] = 1;
            fibonucchi[1] = 1;
            for (int i = 2; i < fibonucchi.Length; i++) fibonucchi[i] = fibonucchi[i - 1] + fibonucchi[i - 2];
            for (int h = 1; h < H + 1; h++)
            {
                for (int w = 0; w < W; w++)
                {
                    long X = Fibonucchi(w - 1) * Fibonucchi(W - w - 1);
                    long Y = Fibonucchi(w) * Fibonucchi(W - w - 1);
                    long Z = Fibonucchi(w) * Fibonucchi(W - w - 2);
                    dp[w, h] = (X * DP(w - 1, h - 1) + Y * DP(w, h - 1) + Z * DP(w + 1, h - 1)) % div;
                }
            }

            long ans = dp[K - 1, H];
            Console.WriteLine(ans);
            Console.ReadKey();
        }

        static long Fibonucchi(int w)
        {
            if (w < 0) return 0;
            return fibonucchi[w];
        }

        static long DP(int w, int h)
        {
            if (w < 0 || h < 0) return 0;
            else if (w >= dp.GetLength(0) || h >= dp.GetLength(1)) return 0;
            return dp[w, h];
        }
    }
}