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()

base = [[0] * W for _ in range(W)]

for bit in range(2 ** (W - 1)):
    temp = [(bit & (1 << i) != 0) for i in range(W - 1)]
    before = False
    is_ok = True
    for line in temp:
        if line:
            if before:
                is_ok = False
                break
            else:
                before = True
        else:
            before = False
    
    if is_ok:
        before = False
        for i, line in enumerate(temp):
            if line:
                base[i][i + 1] += 1
                base[i + 1][i] += 1
                before = True
            else:
                if not before:
                    base[i][i] += 1

                before = False

        if not temp[-1]:
            base[-1][-1] += 1
dp = [[0] * W for _ in range(H + 1)]
dp[0][0] = 1
for i in range(1, H + 1, 1):
    for j in range(W):
        for k in range(W):
            dp[i][j] = (dp[i][j] + dp[i - 1][k] * base[k][j]) % mod

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