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)

h,w,k = gets.split.map(&:to_i)

def hoge(digit, from_w, to_w)
  from = ("0" * digit).to_i(2)
  to = ("1" * digit).to_i(2)
  count = 0
  (from..to).each do |i|
    bit = i.to_s(2).rjust(digit, "0")
    next if bit.include?("11")
    next if (from_w - to_w).abs >= 2
    right = from_w == bit.length ? nil : bit[from_w]
    left = from_w == 0 ? nil : bit[from_w - 1]
    direction = if to_w < from_w
                  count += 1 if left == "1" && right != "1"
    elsif to_w == from_w
                  count += 1 if left != "1" && right != "1"
    else
                  count += 1 if left != "1" && right == "1"
    end
  end
  count
end

dp = []

h.times do |i|
  dp[i] ||= []
  w.times do |j|
    dp[i][j] ||= 0
    if i == 0
      dp[i][j] = hoge(w-1, 0, j)
    else
      if j == 0
        dp[i][j] += (hoge(w-1, j, j) * (dp[i-1][j] || 0))
        dp[i][j] += (hoge(w-1, j+1, j) * (dp[i-1][j+1] || 0))
      elsif j == w
        dp[i][j] += (hoge(w-1, j-1, j) * (dp[i-1][j-1] || 0))
        dp[i][j] += (hoge(w-1, j, j) * (dp[i-1][j] || 0))
      else
        dp[i][j] += (hoge(w-1, j-1, j) * (dp[i-1][j-1] || 0))
        dp[i][j] += (hoge(w-1, j, j) * (dp[i-1][j] || 0))
        dp[i][j] += (hoge(w-1, j+1, j) * (dp[i-1][j+1] || 0))
      end
    end
  end
end

puts dp[-1][k-1] % 1000000007