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>
#include <vector>
#include <algorithm>
#include <cmath>
#include <numeric>
using namespace std;
const int mod = 1000000007;

int main(){
  int h, w, k;
  cin >> h >> w >> k;
  vector<vector<int> > dp(h + 1, vector<int>(w, 0));
  dp[0][0] = 1;
  for (int i = 0; i < h; i++){
    for (int j = 0; j < w; j++){
      for (int k = 0; k < (1 << (w - 1)); k++){
        bool ok = true;
        for (int l = 0; l < w - 2; l++){
          if (((k >> l) & 1) && ((k >> (l + 1)) & 1))
            ok = false;
        }
        if (ok){
          if (j >= 1  && ((k >> (j - 1)) & 1)){
            dp[i + 1][j - 1] += dp[i][j];
            dp[i + 1][j - 1] %= mod;
          }
          else if (j < w - 1 && ((k >> j) & 1)){
            dp[i + 1][j + 1] += dp[i][j];
            dp[i + 1][j + 1] %= mod;
          }
          else{
            dp[i + 1][j] += dp[i][j];
            dp[i + 1][j] %= mod;
          }
        }
      }
    }
  }
  cout << dp[h][k - 1] << endl;
  return 0;
}