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)

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

if W == 1:
    print(1)
    exit()

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

for i in range(H):
    for j in range(W):
        dp[i][j] %= MOD
        here = dp[i][j]
        for d in range(2 ** (W - 1)):
            d = format(d, '0' + str(W - 1) + 'b')
            if not d.count('11'):
                if j == 0:
                    if d[j] == '1':
                        dp[i + 1][j + 1] += here
                    else:
                        dp[i + 1][j] += here
                elif j == W - 1:
                    if d[j - 1] == '1':
                        dp[i + 1][j - 1] += here
                    else:
                        dp[i + 1][j] += here
                else:
                    if d[j - 1] == '1':
                        dp[i + 1][j - 1] += here
                    elif d[j] == '1':
                        dp[i + 1][j + 1] += here
                    else:
                        dp[i + 1][j] += here

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