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: Accepted)

import itertools


def amida_patterns(w):
    # (0, 1, 2, ... , w - 1) -> ?
    ret = list()
    for bars in itertools.product([0, 1], repeat=w-1):
        if sum([b * bb for b, bb in zip(bars[1:], bars[:-1])]) > 0:
            continue
        location = list(range(w))
        for i in range(w):
            if i > 0 and bars[i - 1] == 1:
                location[i] = i - 1
            elif i < w - 1 and bars[i] == 1:
                location[i] = i + 1
        ret.append(location)
    return ret


def main():
    MOD = 10**9 + 7
    H, W, K = list(map(int, input().split(' ')))
    ap_list = amida_patterns(W)
    dp = [[0 for _ in range(W)] for _ in range(H + 1)]
    dp[0][0] = 1
    for h in range(1, H + 1):
        for a in ap_list:
            for w in range(W):
                dp[h][a[w]] += dp[h - 1][w]
                dp[h][a[w]] %= MOD
    print(dp[H][K - 1])


if __name__ == '__main__':
    main()