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())

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

#dp[i][j]：左からj番目の縦棒で上からiの地点を訪れる場合の数
dp = [[0 for _ in range(W)] for _ in range(H+1)]
#最初は一番左上
dp[0][0] = 1

MOD = 10**9+7
for i in range(H):
    # 横棒のパターンを全探索。横棒は二つの縦棒の間にできるので、W-1箇所入れられるのでそれぞれについてあるORない
    # 1ならそこに横棒がある
    for yoko in range(2**(W-1)):
        if "11" in bin(yoko):
            continue
        
        yoko = bin(yoko)[2:].zfill(W-1)

        for j in range(W):
            #dp[i][j]からどこに行くか考える
            #print(j, len(yoko))
            # 左に行く横棒がある場合
            if j != 0 and  yoko[j-1] == "1":
                dp[i+1][j-1] += dp[i][j]
                dp[i+1][j-1] = dp[i+1][j-1] % MOD

            # 右に行く横棒がある場合
            
            elif j != W-1 and yoko[j] == "1":
                dp[i+1][j+1] += dp[i][j]
                dp[i+1][j+1] = dp[i+1][j+1] % MOD

            # そのまま降りる（横棒がない）場合
            else:
                dp[i+1][j] += dp[i][j]
                dp[i+1][j] = dp[i+1][j] % MOD


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