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)

str = gets.split
H,W,K = str[0].to_i, str[1].to_i, str[2].to_i

n = []

n[0] = 1+0
n[1] = 1+0
n[2] = 1+1
n[3] = 1+2
n[4] = 1+3+1
n[5] = 1+4+3
n[6] = 1+5+6+1
n[7] = 1+6+10+4

a = Array.new(H+1).map{Array.new(W+1)}

a[0][1] = 1
for x in 2..W
  a[0][x] = 0
end

for x in 1..H
  for y in 1..W
    a[x][y] = a[x-1][y] * (n[y-1] * n[W-y])
  end
  for y in 2..W
    a[x][y-1] += a[x-1][y] * (n[y-2] * n[W-y])
  end
  for y in 1...W
    a[x][y+1] += a[x-1][y] * (n[y-1] * n[W-y-1])
  end
  for y in 1..W
    a[x][y] = a[x][y] % 1000000007
  end
end

puts(a[H][K])