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 = list(map(int, input().split()))
mod=int(1e9+7)

memo = [[0]*W for _ in range(H+1)]
memo[0][0]=1
#memo[h][w] = h行目のw列の場合の数

from itertools import product
def one_row(W):
    one_row_amida={i:{i:0 for i in range(W)} for i in range(W)}

    for p in product([0,1], repeat=W-1):
        if any([p[i]==1 and p[i+1]==1 for i in range(W-2)]):
            continue
        after = [i for i in range(W)]
        for i in range(W-1):
            if p[i]:
                after[i], after[i+1] = after[i+1], after[i]
        for i, a in enumerate(after):
            one_row_amida[i][a] += 1
    return one_row_amida

one_row_amida = one_row(W)

for h in range(H):
    for w in range(W):
        for j in range(W):
            memo[h+1][j] += memo[h][w] * one_row_amida[w][j]
    memo[h+1] = [x%mod for x in memo[h+1]]

answer = memo[H][K-1]

print(answer)