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_row?(row)
  (row.size-1).times do |j|
    return false if row[j]=='1' and row[j+1]=='1'
  end
  true
end

def valid_rows(w)
  (2**(w-1)).times.map do |i|
    sprintf("%0#{w-1}b0", i)  # last bit is always zero
  end.select do |row|
    is_valid_row?(row)
  end
end

$dp_count_valid_rows = {}

def count_valid_rows(w, ini, dx)
  key = "#{ini},#{dx}"
  if $dp_count_valid_rows[key]
    return $dp_count_valid_rows[key]
  end

  if dx == 0
    ans = valid_rows(w).count do |row|
      row[ini] == '0' and row[ini-1] == '0'
    end
  elsif dx == 1
    ans = valid_rows(w).count do |row|
      row[ini] == '1'
    end
  elsif dx == -1
    ans = valid_rows(w).count do |row|
      row[ini-1] == '1'
    end
  else
    ans = 0
  end
  $dp_count_valid_rows[key] = ans
  return $dp_count_valid_rows[key]
end

$dp_count_correct_amida = {}

D = 1000000007
  

def count_correct_amida(h, w, ini, dest)
  key = "#{h},#{ini},#{dest}"
  if $dp_count_correct_amida[key]
    return $dp_count_correct_amida[key]
  end

  if ini >= w or ini < 0
    ans = 0
  elsif h == 0
    ans = (dest == ini ? 1 : 0)
  else
    n0 = count_valid_rows(w, ini, 0) * count_correct_amida(h-1, w, ini, dest)
    n1 = count_valid_rows(w, ini, 1) * count_correct_amida(h-1, w, ini+1, dest)
    n2 = count_valid_rows(w, ini, -1) * count_correct_amida(h-1, w, ini-1, dest)
    ans = (n0 + n1 + n2) % D
  end
  $dp_count_correct_amida[key] = ans
  $dp_count_correct_amida[key]
end

H,W,K = readline.chomp.split.map(&:to_i)
puts count_correct_amida(H, W, 0, K-1)