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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <iostream>
using namespace std;

long long fib(int k){
    if (k <= -2) return 0LL;
    if (k <= 0) return 1LL;
    return fib(k-1) + fib(k-2);
}

int main(){
    const long long MOD = 1e9+7;
    const int MAX_H = 100;
    const int MAX_W = 8;
    int H, W, K;
    long long dp[MAX_H + 10][MAX_W + 10];

    cin >> H >> W >> K;
    for (int i = 0; i <= H; ++i){
        for (int j = 1; j <= W; ++j){
            if (i == 0 && j == 1) dp[i][j] = 1LL;
            else dp[i][j] = 0LL;
        }
    }
    
    long long tmp;
    for (int i = 1; i <= H; ++i){
        for (int j = 1; j <= W; ++j){
            
            if (j - 1 >= 1){
                tmp = dp[i-1][j-1] * fib((j-2)-1) % MOD;
                tmp = tmp * fib(W-(j+1)) % MOD;
                dp[i][j] = (dp[i][j] + tmp) % MOD;
            }
            
            tmp = dp[i-1][j] * fib((j-1)-1) % MOD;
            tmp = tmp * fib(W-(j+1)) % MOD;
            dp[i][j] = (dp[i][j] + tmp) % MOD;
            
            if (j + 1 <= W){
                tmp = dp[i-1][j+1] * fib((j-1)-1) % MOD;
                tmp = tmp * fib(W-(j+2)) % MOD;
                dp[i][j] = (dp[i][j] + tmp) % MOD;
            }
        }
    }
    cout << dp[H][K] << endl;
}