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 <set>
#include <algorithm>
#include <math.h>
#include <string.h>
#include <map>
#include <numeric>
#include <random>

using namespace std;

const int MOD = 1000000007;
typedef long long ll;

int main()
{
  cin.tie(0);
  ios::sync_with_stdio(false);
  int H,W,K;
  cin>>H>>W>>K;
  vector<vector <ll> > dp(H+1, vector<ll>(W,0));
  dp[0][0]=1ll;
  for(int i=0;i<H;i++){
    for(int j=0;j<W;j++){
      for (int bit = 0; bit < 1<<(W-1); ++bit) {
        bool ok = true;
        for (int k = 0; k < W-2; ++k) {
          if ((bit & (1<<k)) && (bit & (1<<(k+1)))) ok = false;
        }
        if (!ok) continue;
        int nj = j;
        if (bit & (1<<j)) nj = j+1;
        else if (j > 0 && (bit & (1<<(j-1)))) nj = j-1;
        dp[i+1][nj] += dp[i][j];
        dp[i+1][nj] %= MOD;
      }
    }
  }
  cout<<dp[H][K-1]<<endl;
}