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.Text;

namespace AtTest.ABC_113
{
    class D
    {
        static void Main(string[] args)
        {
            Method(args);
            Console.ReadLine();
        }

        static void Method(string[] args)
        {
            int[] hwk = ReadInts();
            int h = hwk[0];
            int w = hwk[1];
            int k = hwk[2] - 1;
            long mask = 1000000000 + 7;
            var rowPats = new long[6] { 2, 3, 5, 8, 13, 21 };
            var dp = new long[h + 1, w];
            dp[0, 0] = 1;
            for (int i = 1; i <= h; i++)
            {
                for (int j = 0; j < w; j++)
                {
                    //まうえ
                    dp[i, j] = dp[i - 1, j];
                    dp[i, j] %= mask;
                    if (j >= 2)
                    {
                        dp[i,j]*= rowPats[j - 2];
                        dp[i, j] %= mask;
                    }
                    if (j + 3 <= w)
                    {
                        dp[i, j] *= rowPats[w - j - 3];
                        dp[i, j] %= mask;
                    }
                    dp[i, j] %= mask;

                    if (j < w - 1)//右
                    {
                        long right = dp[i - 1, j + 1];

                        if (j + 1 >= 3)
                        {
                            right *= rowPats[j - 2];
                            right %= mask;
                        }
                        if (j + 4 <= w)
                        {
                            right *= rowPats[w - j - 4];
                            right %= mask;
                        }
                        dp[i, j] += right;
                        dp[i, j] %= mask;
                    }
                    if (0 < j)//left
                    {
                        long left = dp[i - 1, j - 1];
                        if (j - 3 >= 0)
                        {
                            left *= rowPats[j - 3];
                            left %= mask;
                        }
                        if (j + 3 <= w)
                        {
                            left *= rowPats[w - j - 3];
                            left %= mask;
                        }
                        dp[i, j] += left;
                        dp[i, j] %= mask;
                    }
                    //Console.Write(dp[i, j] + " ");
                }
                //Console.WriteLine();
            }
            Console.WriteLine(dp[h, k]);
        }

        private static string Read() { return Console.ReadLine(); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
    }
}