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


def violates(w_flags):
    prev_flag = w_flags[0]
    for flag in w_flags[1:]:
        if flag and prev_flag:
            return True
        prev_flag = flag
    return False


H, W, K = [int(elem) for elem in input().split()]
MOD = 10**9 + 7

if W == 1:
    print(1)
    exit(0)

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

for h in range(1, H + 1):
    for w_flags in product((False, True), repeat=W-1):
        if violates(w_flags):
            continue
        for w in range(W):
            if w == 0:
                if w_flags[0]:
                    dp[h][0] += dp[h - 1][1]
                else:
                    dp[h][0] += dp[h - 1][0]
            elif w == W - 1:
                if w_flags[W - 2]:
                    dp[h][W - 1] += dp[h - 1][W - 2]
                else:
                    dp[h][W - 1] += dp[h - 1][W - 1]
            else:
                if w_flags[w - 1] == True:
                    dp[h][w] += dp[h - 1][w - 1]
                elif w_flags[w] == True:
                    dp[h][w] += dp[h - 1][w + 1]
                else:
                    dp[h][w] += dp[h - 1][w]
            dp[h][w] %= MOD

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