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)

from itertools import product
H, W, K = map(int, input().split())
MOD = 10 ** 9 + 7

dp = [[0] * W for i in range(H + 1)]
dp[0][0] = 1


def calc(w):
    if w <= 0:
        return 1
    ret = 0
    for pattern in product([0, 1], repeat=w):
        pattern = ''.join(list(map(str, pattern)))
        if '11' not in pattern:
            ret += 1
    return ret


for h in range(H):
    for w in range(W):
        dp[h + 1][w] = (dp[h + 1][w] + dp[h][w] * calc(w - 1) * calc(W - w - 2)) % MOD
        if w + 1 < W:
            dp[h + 1][w + 1] = (dp[h + 1][w + 1] + dp[h][w] * calc(w - 1) * calc(W - w - 3) % MOD)
        if w - 1 >= 0:
            dp[h + 1][w - 1] = (dp[h + 1][w - 1] + dp[h][w] * calc(w - 2) * calc(W - w - 2) % MOD)

print(dp[H][K - 1] % MOD)