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)

M = 1_000_000_007

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

dir = Array.new(W) { Hash.new(0) }
fin = 1 << (W - 1)
(0...fin)
    .map{|i| i.to_s(2)}
    .select{|s| s !~ /11/}
    .map {|s| ("%0#{W}d0" % s).split('').map(&:to_i)}
    .each do |a| 
      a.each_cons(2).with_index do |(left, right), i|
        if left == 1
          dir[i][i - 1] += 1
        elsif right == 1
          dir[i][i + 1] += 1
        else  
          dir[i][i] += 1
        end
      end
    end

x = Array.new(W, 0)
x[0] = 1
y = Array.new(W, 0)
H.times do
  dir.each.with_index do |h, i|
    h.each do |j, c|
      y[j] = (y[j] + x[i] * c) % M  
    end 
  end
  x, y = y, x
  y = Array.new(W, 0)
end

puts x[K - 1]