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 combinations
mod = 10**9 + 7
h, w, k = map(int, input().split())
edge = [[0]*w for i in range(w)]

for r in range(w//2+1):
    for c in combinations(range(w-1),r):
        # 横線が隣接する場合を排除
        flag = True
        if len(c) > 1:
            for i in range(len(c)-1):
                if c[i+1] - c[i] == 1:
                    flag = False
                    break
        if not flag:continue
        for i in range(w):
            if i in c: continue
            if i-1 in c:
                edge[i-1][i] += 1
                edge[i][i-1] += 1
            else:
                edge[i][i] += 1

# DP[i][j] == i行目まで見た時に1がjの位置にある場合の数
DP = [[0]*w for i in range(h+1)]
DP[0][0] = 1
for i in range(h):
    for j in range(w):
        for l in range(w):
            DP[i+1][j] = (DP[i+1][j] + DP[i][l]*edge[l][j]) % mod
#print(DP)
print(DP[-1][k-1])