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)
c = Array.new(w-1,0)
a = Array.new(w-1,0)
s = 0
for i in 0..2**(w-1)-1
    (w-1).times do |j|
        a[j] = i[j]
    end
    f = true
    for j in 0..w-3
        if a[j] == 1 && a[j+1] == 1
            f = false
            break
        end
    end
    if f
        s += 1
        (w-1).times do |j|
            c[j] += a[j]
        end
    end
end
ans = Array.new(h+1).map{Array.new(w,0)}
ans[0][0] = 1
$mod = 10**9+7
for i in 1..h
    for j in 0..w-1
        t = 0
        if j != 0
            t += c[j-1]
            ans[i][j-1] += ans[i-1][j]*c[j-1]
        end
        if j != w-1
            t += c[j]
            ans[i][j+1] += ans[i-1][j]*c[j]
        end
        ans[i][j] += ans[i-1][j]*(s-t)
        ans[i][j] %= $mod
    end
end
puts ans[h][k-1]%$mod