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)
MOD = 10 ** 9 + 7
dp = Array.new(h+2){Array.new(w+1,0)}
dp[0][1] = 1
h.times do |y|
  (2**(w-1)).times do |i|
    cross = i.to_s(2).rjust(w-1,"0")
    valid = true
    (w-2).times do |i|
      valid = false if cross[i] == "1" && cross[i+1] == "1"
    end
    if valid
      1.upto(w) do |x|
        if x == 1
          if cross[0] == "1"
            dp[y+1][x] += dp[y][x+1]
          else
            dp[y+1][x] += dp[y][x]
          end
        elsif x == w
          if cross[w-2] == "1"
            dp[y+1][x] += dp[y][x-1]
          else
            dp[y+1][x] += dp[y][x]
          end
        else
          if cross[x-2] == "1"
            dp[y+1][x] += dp[y][x-1]
          elsif cross[x-1] == "1"
            dp[y+1][x] += dp[y][x+1]
          else
            dp[y+1][x] += dp[y][x]
          end
        end
        dp[y+1][x] = dp[y+1][x] % MOD
      end
    end
  end
end
p dp[h][k]