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 (GCC 5.4.1) (judge verdict: Accepted)

#include <stdio.h>

#define MOD 1000000007

int main() {
  int h, w, k;
  scanf("%d%d%d", &h, &w, &k);
  int dp[h+1][w];
  for (int i=0; i<h+1; i++) {
    for (int j=0; j<w; j++) {
      dp[i][j] = 0;
    }
  }
  dp[0][0] = 1;
  for (int i=1; i<h+1; i++) {
    for (int k=0; k<(1<<(w-1)); k++) {
      int check = 1;
      for (int l=0; l<w-2; l++) {
        if (((k>>l)&3) == 3) {
          check = 0;
          break;
        }
      }
      if (check == 0) continue;
      for (int j=0; j<w; j++) {
        if (j<w-1 && ((k>>j)&1) == 1) {
          dp[i][j+1] += dp[i-1][j];
          dp[i][j+1] %= MOD;
        }else if (j>0 && ((k>>(j-1))&1) == 1) {
          dp[i][j-1] += dp[i-1][j];
          dp[i][j-1] %= MOD;
        }else {
          dp[i][j] += dp[i-1][j];
          dp[i][j] %= MOD;
        }
      }
    }
  }
  printf("%d\n", dp[h][k-1]);
  return 0;
}