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 itertools

def amida_counter(H_i_amida_list):
    counter = 0

    for amida_bin in range(2**len(H_i_amida_list)):
        if (("11" in format(amida_bin, "b")) == False):
            counter += 1

    return counter

def amida_number(left_amida, right_amida):
    left_counter = amida_counter(left_amida) if(len(left_amida) > 0) else 1
    right_counter = amida_counter(right_amida) if(len(right_amida) > 0) else 1

    return (left_counter * right_counter)

def X_amida_number(W, W_i):
    x_amida = "0" * (W - 1)
    left_x_amida = x_amida[:(W_i-2)] if(W_i-2 >= 0) else ""
    right_x_amida = x_amida[(W_i+1):]

    return amida_number(left_x_amida, right_x_amida)

def Y_amida_number(W, W_i):
    y_amida = "0" * (W - 1)
    left_y_amida = y_amida[:W_i-1] if(W_i-1 >= 0) else ""
    right_y_amida = y_amida[W_i+1:]

    return amida_number(left_y_amida, right_y_amida)

def Z_amida_number(W, W_i):
    z_amida = "0" * (W - 1)
    left_z_amida = z_amida[:W_i-1] if(W_i-1 >= 0) else ""
    right_z_amida = z_amida[W_i+2:]

    return amida_number(left_z_amida, right_z_amida)

H, W, K = map(int, input().split())
seed_dp_array = [0 for i in range(W)]
dp_array = [list(seed_dp_array) for i in range(H + 1)]
dp_array[0][0] = 1

for H_i, W_i in itertools.product(range(1, H + 1), range(W)):
    if (W_i == 0):
        if(W == 1):
            dp_array[H_i][0] = dp_array[H_i - 1][W_i] * Y_amida_number(W, W_i)
        else:
            dp_array[H_i][0] = dp_array[H_i - 1][W_i] * Y_amida_number(W, W_i) + dp_array[H_i - 1][W_i + 1] * Z_amida_number(W, W_i)
    elif (W_i == (W - 1)):
        dp_array[H_i][W - 1] = dp_array[H_i - 1][W_i - 1] * X_amida_number(W, W_i) + dp_array[H_i - 1][W_i] * Y_amida_number(W, W_i)
    else:
        dp_array[H_i][W_i] = dp_array[H_i - 1][W_i - 1] * X_amida_number(W, W_i) + dp_array[H_i - 1][W_i] * Y_amida_number(W, W_i) + dp_array[H_i - 1][W_i + 1] * Z_amida_number(W, W_i)

print(dp_array[H][K - 1] % 1000000007)