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)

def main():
    import sys
    input = sys.stdin.readline
    sys.setrecursionlimit(10**7)
    from collections import Counter, deque
    #from collections import defaultdict
    from itertools import combinations, permutations, accumulate, groupby
    #from itertools import product
    from bisect import bisect_left,bisect_right
    from heapq import heapify, heappop, heappush
    from math import floor, ceil
    #from operator import itemgetter

    #inf = 10**17
    mod = 10**9 + 7

    h,w,k = map(int, input().split())
    dp = [[0]*w for _ in range(h+1)]
    dp[0][0] = 1

    pattern = []
    if w > 2:
        for i in range(1<<(w-1)):
            for j in range(w-2):
                if ((i>>j) & 1) and ((i>>(j+1)) & 1):
                    break
            else:
                pattern.append(i)
    else:
        print(1)
        exit()
    
    for i in range(h):
        for j in range(w):
            for p in pattern:
                if j == 0:
                    if p & 1:
                        dp[i+1][j+1] += dp[i][j] % mod
                    else:
                        dp[i+1][j] += dp[i][j] % mod
                elif j == w-1:
                    if (p>>(w-2)) & 1:
                        dp[i+1][j-1] += dp[i][j] % mod
                    else:
                        dp[i+1][j] += dp[i][j] % mod
                else:
                    if (p>>(j-1)) & 1:
                        dp[i+1][j-1] += dp[i][j] % mod
                    elif (p>>j) & 1:
                        dp[i+1][j+1] += dp[i][j] % mod
                    else:
                        dp[i+1][j] += dp[i][j] % mod
    print(dp[-1][k-1] % mod)


if __name__ == '__main__':
    main()