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 <bits/stdc++.h>
using namespace std;

int main() {
  int h, w, k;
  cin>>h>>w>>k;
  long long mod = 1e9 + 7;

  long long dp1[3][w];
  dp1[0][0] = dp1[0][1] = 1;
  dp1[0][2] = dp1[1][2] = 1;
  dp1[1][1] = dp1[1][0] = 0;
  for (int i=3; i<w; i++) {
    dp1[0][i] = (dp1[0][i-1] + dp1[1][i-1]) % mod;
    dp1[1][i] = dp1[0][i-1];
  }
  for (int i=0; i<w; i++) dp1[2][i] = (dp1[0][i] + dp1[1][i]) % mod;

  long long ans[h+1][w];
  for (int i=0; i<=h; i++) {
    for (int j=0; j<w; j++) {
      ans[i][j] = 0;
    }
  }
  ans[0][0] = 1;
  for (int i=1; i<=h; i++) {
    for (int j=0; j<w; j++) {
      if (j > 0) {
        ans[i][j] += (((ans[i-1][j-1] * dp1[2][j-1]) % mod) * dp1[2][w-j-1]) % mod;
        ans[i][j] %= mod;
      }
      ans[i][j] += (((ans[i-1][j] * dp1[2][j]) % mod) * dp1[2][w-j-1]) % mod;
      ans[i][j] %= mod;
      if (j < w - 1) {
        ans[i][j] += (((ans[i-1][j+1] * dp1[2][j]) % mod) * dp1[2][w-j-2]) % mod;
        ans[i][j] %= mod;
      }
    }
  }
  cout<<ans[h][k-1]<<endl;
}