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 (2.7.1) (judge verdict: Wrong Answer)

MOD = 10**9+7
h, w, k = gets.split.map &:to_i

if w == 1
  p 1
  exit
end

dp = [[0] * w]
dp[0][0] = 1
h.times do |i|
  dp[i+1] = []
  dp[i+1][0] = ((dp[i][0] << [w-2, 0].max) + (dp[i][1] << [w-3, 0].max) ) % MOD
  (1..w-2).each do |j|
    dp[i+1][j] = ((dp[i][j] << [j-1, 0].max + [w-j-2, 0].max) + (dp[i][j-1] << [j-2, 0].max + [w-j-2, 0].max) + (dp[i][j+1] << [j-1, 0].max + [w-j-3, 0].max)) % MOD
  end
  dp[i+1][w-1] = ((dp[i][w-1] << [w-2, 0].max) + (dp[i][w-2] << [w-3, 0].max)) % MOD
end
p dp[h][k-1] % MOD