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 (Clang 3.8.0) (judge verdict: Accepted)

#include <cstdio>
#include <cstdint>
#include <vector>
#include <algorithm>

constexpr intmax_t MOD = 1e9+7;

int main() {
  size_t H, W, K;
  scanf("%zu %zu %zu", &H, &W, &K);

  std::vector<std::vector<intmax_t>> dp(H+1, std::vector<intmax_t>(W));
  dp[0][0] = 1;

  for (size_t i = 0; i < H; ++i) {
    for (size_t j = 0; j < W; ++j) {
      for (int k = 0; k < (1 << (W-1)); ++k) {
        if (k & (k >> 1)) continue;
        if (j+1 < W && k >> j & 1) {
          (dp[i+1][j+1] += dp[i][j]) %= MOD;
        } else if (j > 0 && k >> (j-1) & 1) {
          (dp[i+1][j-1] += dp[i][j]) %= MOD;
        } else {
          (dp[i+1][j] += dp[i][j]) %= MOD;
        }
      }
    }
  }
  printf("%jd\n", dp[H][K-1]);
}