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 PyPy3 (7.3.0) (judge verdict: Accepted)

H, W, K = map(int, input().split())
MOD, p, cnt = 10**9+7, [0]*W, 0
for bit in range(1<<(W-1)):
    s0 = bit&1
    flg=True
    for w in range(1, W-1):
        if (bit>>w)&1 and s0 == 1:flg=False; break
        s0 = (bit>>w)&1
    if flg:
        cnt += 1
        for w in range(W - 1):
            if (bit >> w)&1:p[w] += 1
dp = [[0]*W for _ in range(H+1)]
dp[0][0] = 1
for h in range(H):
    for w in range(W):
        if w:
            dp[h+1][w-1] += dp[h][w]*p[w-1]
            dp[h+1][w-1] %= MOD
        if w-(W - 1):
            dp[h+1][w+1] += dp[h][w]*p[w]
            dp[h+1][w+1] %= MOD
        dp[h+1][w]+=dp[h][w]*(cnt-p[w-1]-p[w])
        dp[h+1][w]%=MOD
print(dp[H][K-1])