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 9.2.1) (judge verdict: Accepted)

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

int main() {
  int H,W,K;
  cin>>H>>W>>K;
  K--;
  vector<int> p(W);
  for(int i=0;i<W;i++){
    p[i]=i;
  }
  vector<vector<int>> q;
  for(int i=0;i<(1<<(W-1));i++){
    vector<int> r;
    int S=0,A=0;
    vector<int> s=p;
    for(int j=0;j<W-1;j++){
      if(i&(1<<j)){
        r.push_back(j);
        S++;
      }
    }
    for(int j=0;j<S;j++){
      swap(s[r[j]],s[r[j]+1]);
      if(j==0){
        continue;
      }
      else if(r[j]-r[j-1]==1){
        A=1;
        break;
      }
    }
    if(A){
      continue;
    }
    else{
      q.push_back(s);
    }
  }
  int N=q.size();
  vector<vector<int64_t>> dp(H+1,vector<int64_t>(W));
  dp[0][0]=1;
  int64_t Q=1000000007;
  for(int i=0;i<H;i++){
    for(int j=0;j<N;j++){
      for(int k=0;k<W;k++){
        dp[i+1][q[j][k]]+=dp[i][k];
        dp[i+1][q[j][k]]%=Q;
      }
    }
  }
  cout<<dp[H][K]<<endl;
  
}