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)

import numpy as np
H, W, K = map(int, input().split())

if W == 1:
    print (1)
    exit()
MOD = 10 ** 9 + 7

#dp-tableの作成
lst = [[0] * (W) for i in range(H+2)]
dp = np.array(lst)
dp[0][0] = 1

def check(h): #h番目のX, Y, Zを検索
    count = 0
    for i in range(2**(W-1)):
        flag2 = True
        flag = False #一つ前が橋のとき、フラグをTrueにする
        for j in range(W-1):
            if flag2: #どこかで連続橋になったらそれ以降計算しない
                # print (j, end = ' ')
                if (i >> j) & 1: #1を橋とする
                    if flag: #連続で橋の時
                        count += 1
                        # print ('A', end = ' ')
                        flag = True
                        flag2 = False
                        break
                    else: #一つ前が橋では無い時-->ここに橋をかける
                        dp[H+1][j] += dp[h-1][j+1]%MOD
                        dp[H+1][j+1] += dp[h-1][j]%MOD
                        flag = True
                else:
                    if j == W-2: #最後の一本が橋でない時
                        dp[H+1][j+1] = dp[h-1][j+1]%MOD
                    if not flag: #連続で橋がかからない時、まっすぐ下ろす
                        dp[H+1][j] = dp[h-1][j]%MOD
                    else:
                        pass
                    flag = False
        if flag2: #最後まで条件を満たしていたら、H+1に記録していたものを通常の場所に移す
            dp[h][:] += dp[H+1][:]%MOD
            dp[h][:] = dp[h][:]%MOD
            # print (dp[H+1])
        dp[H+1][:] = 0
        # print ()
    # print (i)
    # print (count)
    return 0;

for i in range(H):
    check(i+1)

# print (dp)
print (dp[H][K-1]%MOD)