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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

def Calc(M):
    meta = (1,1,2,3,5,8,13,21)
    result = 1
    for i in M:
        result *= meta[i[0] - 1] * meta[W - i[0] - i[1]]
    return result % 1000000007

def Map(X, Y, M):
    if Y == H:
        if X == K + 1:
            M.append((X - 1, 1))
            return Calc(M)
        if X == K:
            M.append((X, 0))
            return Calc(M)
        if X == K - 1:
            M.append((X, 1))
            return Calc(M)
        return 0
    
    XnM = M.copy()
    XnM.append((X, 0))
    result = Map(X, Y + 1, XnM)

    if X > 1:
        XmM = M.copy()
        XmM.append((X - 1, 1))
        result += Map(X - 1, Y + 1, XmM)

    if X < W:    
        XpM = M.copy()
        XpM.append((X, 1))
        result += Map(X + 1, Y + 1, XpM)

    return result % 1000000007

ls = input().strip().split(' ')
H = int(ls[0])
W = int(ls[1])
K = int(ls[2])

print(Map(1, 1, []))