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 Ruby (2.3.3) (judge verdict: Accepted)

MOD = 1_000_000_007
H, W, K = gets.split.map(&:to_i)

if W == 1
  puts 1
else
  # Array dpのインデックスはすべて、問題文の数-1になっていることに注意
  
  # (0番目から数えて)8番目までのフィボナッチ数を作っておく
  # fib[i]は、i本の縦棒があるときの横棒の引き方の総数
  # 縦棒のインデックスがkなら、fib[k+1]が対応
  fib = [1, 1, 2, 3, 5, 8, 13, 21, 34]
  
  dp = Array.new(H).map{Array.new(W).map{Array.new(W, 0)}}
  
  # H = 1の場合
  W.times do |w|
    dp[0][w][0] = fib[w]
    dp[0][w][1] = fib[w-1] if w > 0
  end
  
  # 高さが1つ少ないときのパターン数が分かれば、以下で求められる
  # そのまま降りてくるパターン数 × その前後に横棒を引かない横棒の引き方総数
  # + 1つ前から移動してくる × 1つ前に横棒を引く横棒の引き方総数
  # + 1つ後ろから移動してくる × 1つ後ろに横棒を引く横棒の引き方総数
  1.upto(H-1) do |h|
    W.times do |w|
      0.upto(w) do |k|
        if k == 0
          dp[h][w][k] = (fib[W-1] * dp[h-1][w][0] % MOD) + (fib[W-2] * dp[h-1][w][1] % MOD)
          dp[h][w][k] %= MOD
        elsif k == W-1
          dp[h][w][k] = (fib[W-1] * dp[h-1][w][W-1] % MOD) + (fib[W-2] * dp[h-1][w][W-2] % MOD)
          dp[h][w][k] %= MOD
        else
          dp[h][w][k] = (fib[k] * fib[W-k-1] * dp[h-1][w][k] % MOD) + (fib[k-1] * fib[W-k-1] * dp[h-1][w][k-1] % MOD) + (fib[k] * fib[W-k-2] * dp[h-1][w][k+1] % MOD)
          dp[h][w][k] %= MOD
        end
      end
    end
  end
  
  puts dp[H-1][W-1][K-1]
end