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)

# coding: utf-8
# Your code here!

import itertools 
from collections import defaultdict

H, W, K = map(int, input().rstrip().split(' '))


w = [[0] * 3 for _ in range(W)]
patterns = itertools.product([0, 1], repeat=W-1)

if W > 1:
    for i in range(W):
        for k, j in enumerate([-1, 0, 1]):
            if i - j < 0 or i - j > W - 1:
                continue
            
            num_use_pattern = 0
            for pattern in itertools.product([0, 1], repeat=W-1):
                if j == -1:
                    if i - 1 < 0: 
                        if pattern[i] == 0:
                            continue
                    else:
                        if not (pattern[i-1] == 0 and pattern[i] == 1):
                            continue
                elif j == 1:
                    if i >= W - 1:
                        if pattern[i-1] == 0:
                             continue
                    else:
                        if not (pattern[i-1] == 1 and pattern[i] == 0):
                            continue
                else:
                    if i - 1 < 0: 
                        if pattern[i] == 1:
                            continue
                    elif i >= W - 1:
                        if pattern[i-1] == 1:
                             continue
                    else:
                        if pattern[i] == 1 or pattern[i-1] == 1:
                            continue
                   
                if sum([pattern[l] * pattern[l-1] for l in range(1, W-1)]) > 0:
                    continue
                
                num_use_pattern += 1
                
            w[i][k] += num_use_pattern
     
    memo = defaultdict(lambda: defaultdict(lambda: -1))
    def recursive(h, k, num_p):
        if 0 > k or k >= W:
            return 0
        if memo[h][k] != -1:
            return memo[h][k] * num_p
        if h == H:
            if k == K - 1:
                return num_p
            else:
                return 0
             
        temp1 = 0
        temp2 = 0
        temp3 = 0
        temp1 = recursive(h + 1, k, w[k][1])
        if k > 0:
            temp2 = recursive(h + 1, k - 1, w[k][2])
        if k <= W - 1:
            temp3 = recursive(h + 1, k + 1, w[k][0])
            
        temp = temp1 + temp2 + temp3
        memo[h][k] = temp
        
        return temp * num_p
     
    print(recursive(0, 0, 1) % 1000000007)
else:
    print(1)