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)

H,W,K = map(int, input().split())

# dp[h][w]
dp = []
for _ in range(H+1):
    dp.append([0] * W)

dp[0][0] = 1

for h in range(1,H+1):
    for w in range(0,W):
        a = 0
        b = 0
        c = 0
        
        Q = W-1
        for bit in range(1<<Q):
            invalid = False
            cur_seq = False
            for i in range(Q):
                on = (bit >> i) & 1
                if on & cur_seq:
                    invalid = True
                    break
                cur_seq = on

            # if there is sequential bar, then the combination is invalid
            if invalid:
                continue

            connected = False
            # if the combination has connection with w then count a or c
            # otherwise, inc b
            for i in range(Q):
                on = (bit >> i) & 1
                if on:
                    x = i
                    y = i+1
                    if x == w-1:
                        connected = True
                        a += 1
                    elif y == w+1:
                        connected = True
                        c += 1
            if not connected:
                b += 1

        cnt = 0
        if a > 0:
            cnt += a * dp[h-1][w-1]
        if b > 0:
            cnt += b * dp[h-1][w]
        if c > 0:
            cnt += c * dp[h-1][w+1]

        dp[h][w] = cnt

print(dp[H][K-1] % 1000000007)