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 math

def main():
    buf = input()
    buflist = buf.split()
    H = int(buflist[0])
    W = int(buflist[1])
    K = int(buflist[2]) - 1 # zero indexing

    pattern_list = []
    create_pattern(pattern_list, [], W-1, 0)
    prob = [1]
    for i in range(W-1):
        prob.append(0)
    for i in range(H):
        new_prob = []
        for j in range(W):
            new_prob.append(0)
        for j in range(W):
            for k in pattern_list:
                if j in k:
                    new_prob[j+1] += prob[j]
                elif j-1 in k:
                    new_prob[j-1] += prob[j]
                else:
                    new_prob[j] += prob[j]
        prob = list(new_prob)
    print(prob[K] % 1000000007)


def create_pattern(pattern_list, pattern, W, C):
    pattern_list.append(pattern)
    if len(pattern) == 0:
        for i in range(W):
            new_pattern = list(pattern)
            new_pattern.append(i)
            create_pattern(pattern_list, new_pattern, W, i)
    else:
        for i in range(C+2, W):
            new_pattern = list(pattern)
            new_pattern.append(i)
            create_pattern(pattern_list, new_pattern, W, i)


def nCr(n, r):
    f = math.factorial
    return f(n) // f(r) // f(n-r)

if __name__ == '__main__':
    main()