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)
MOD = 10 ** 9 + 7

wk = (0...(1<<(W-1))).to_a.select{|n|
  (n[0] == 0 || n[1] == 0) &&
  (n[1] == 0 || n[2] == 0) &&
  (n[2] == 0 || n[3] == 0) &&
  (n[3] == 0 || n[4] == 0) &&
  (n[4] == 0 || n[5] == 0) &&
  (n[5] == 0 || n[6] == 0)
}
t = Array.new(W-1).map.with_index{|m, i|
  wk.count{|n| n[i] == 1}
}
bef = Array.new(W, 0)
bef[0] = 1
H.times{
  nxt = Array.new(W, 0)
  bef.each_with_index{|b, i|
    l = (i==0 ? 0 : t[i-1])
    r = (i==W-1 ? 0 : t[i])
    c = wk.size - l - r
    nxt[i-1] += b*l if i>0
    nxt[i+1] += b*r if i<W-1
    nxt[i] += b*c
  }
  bef = nxt.map{|n| n % MOD}
}

puts bef[K-1]