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;
using System.Collections.Generic;

class Program
{
  static void Main(string[] args)
  {

    var line = Console.ReadLine().Split(' ').Select(Int32.Parse).ToArray();
    var H = line[0];
    var W = line[1];
    var K = line[2];

    var dp = new int[H + 1, W];
    dp[0, 0] = 1;
    for (var hi = 0; hi < H; hi++)
    {
      for (var wi = 0; wi < W; wi++)
      {
        for (var bit = 0; bit < (1 << (W - 1)); bit++)
        {
          var invalid = false;
          for (var i = 0; i < W - 2; i++)
          {
            if ((bit & (1 << i)) != 0 && (bit & (1 << (i + 1))) != 0)
            {
              invalid = true;
            }
          }
          if (invalid) continue;

          var mod = 1000000007;
          if ((bit & (1 << (wi - 1))) != 0 && wi - 1 >= 0)
          {
            dp[hi + 1, wi - 1] += dp[hi, wi];
            dp[hi + 1, wi - 1] %= mod;
          }
          else if ((bit & (1 << wi)) != 0 && wi + 1 < W)
          {
            dp[hi + 1, wi + 1] += dp[hi, wi];
            dp[hi + 1, wi + 1] %= mod;
          }
          else
          {
            dp[hi + 1, wi] += dp[hi, wi];
            dp[hi + 1, wi] %= mod;
          }
        }
      }
    }

    Console.WriteLine(dp[H, K - 1]);
  }
}