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())

# 連続するn区間に橋をかける方法は何通りあるか
# kakekata[b][i] : bが0の時はi区間目に橋をかけない方法 bが1の時はi区間目に橋をかける方法 が何通りあるか
kakekata = [[0] * 10] + [[0] * 10]
kakekata[0][0] = 1
for i in range(8):
    kakekata[0][i + 1] = kakekata[0][i] + kakekata[1][i]
    kakekata[1][i + 1] = kakekata[0][i]


MOD = 10 ** 9 + 7
class modint:
    def __init__(self, x):
        self.x = x.x if isinstance(x, modint) else x % MOD
    def __str__(self): return str(self.x)
    __repr__ = __str__
    def __int__(self): return self.x
    __index__ = __int__
    def __add__(self, other): return modint(self.x + modint(other).x)
    def __sub__(self, other): return modint(self.x - modint(other).x)
    def __mul__(self, other): return modint(self.x * modint(other).x)
    def __pow__(self, other): return modint(pow(self.x, modint(other).x, MOD))
    def __floordiv__(self, other): return modint(self.x * pow(modint(other).x, MOD - 2, MOD))
    def __eq__(self, other): return self.x == modint(other).x
    def __ne__(self, other): return self.x != modint(other).x
    def __inv__(self): return pow(self.x, MOD - 2, MOD)


dp = [[modint(0)] * (W + 2) for h in range(H + 1)]
dp[0][1] += 1
for h in range(H):
    for w in range(1, W + 1):
        # 左に何区間あるか 右に何区間あるか
        l, r = w - 1, W - w

        # そのまま渡す
        dp[h + 1][w] += dp[h][w] * kakekata[0][l] * kakekata[0][r]
        # 右に渡す
        dp[h + 1][w + 1] += dp[h][w] * kakekata[0][l] * kakekata[1][r]
        # 左に渡す
        dp[h + 1][w - 1] += dp[h][w] * kakekata[1][l] * kakekata[0][r]

print(dp[H][K])