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)

def is_valid? bin
  return false if bin.include? "11"
  true
end

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

transition = Array.new(w + 1) { Array.new(w + 1, 0) }
transition[0][0] = 1 if w == 1
(2 ** (w - 1)).times do |i|
  bin = i.to_s(2)
  if is_valid? bin
    bin = bin.rjust(w - 1, '0')
    (w - 1).times do |k|
      if bin[k] == "1"
        transition[k][k + 1] += 1
        transition[k + 1][k] += 1
      elsif k == 0
        transition[0][0] += 1
      elsif bin[k - 1] == "0"
        transition[k][k] += 1
      end
      transition[w - 1][w - 1] += 1 if bin[k] == "0" && k == w - 2
    end
  end
end

dp = Array.new(h + 1) { Array.new(w + 1, 0) }
dp[0][0] = 1

mod = 1000000007
1.upto (h) do |i|
  w.times do |j|
    dp[i][j] = (dp[i - 1][j - 1] * transition[j - 1][j] + dp[i - 1][j] * transition[j][j] + dp[i - 1][j + 1] * transition[j + 1][j]) % mod
  end
end

p dp[h][k - 1]