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.chomp.split.map(&:to_i)
g = []
for i in 0...w
  g[i] = []
  for j in 0...w
    g[i][j] = 0
  end
end
for i in 0...w
  for j in 0...w
    next if j < i-1
    next if j > i+1
    for msk in 0...(1<<(w-1))
      next if (msk & (msk>>1)) > 0
      next if (j==i) && ((msk&(1<<i))>0 || (i>0 && (msk&(1<<(i-1)))>0))
      next if (j==i+1) && (msk&(1<<i))==0
      next if (j==i-1) && (msk&(1<<(i-1)))==0
      g[i][j] += 1
    end
  end
end
dp = []
for i in 0..h
  dp[i] = []
  for j in 0...w
    dp[i][j] = 0
  end
end
MOD = 1000000007
dp[0][0] = 1
for i in 0...h
  for j in 0...w
    for to in 0...w
      dp[i+1][to] += dp[i][j] * g[j][to] % MOD
      dp[i+1][to] %= MOD
    end
  end
end
puts dp[h][k-1]