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;
using vi = vector<int>;
using vb = vector<bool>;
using vc = vector<char>;
using vs = vector<string>;
using vvi = vector<vector<int>>;
using vvc = vector<vector<char>>;
using pii = pair<int, int>;
#define fix10 cout << fixed << setprecision(10);
#define rep(i,n) for(int i=0; i<(int)(n);i++)
#define MOD 1000000007

int main(){
  int h,w,k;
  cin >> h >> w >> k;
  vector<vector<long>> dp(h+1,vector<long>(w,0));
  dp.at(0).at(0) = 1;
  long data[8] = {1,1,2,3,5,8,13,21};
  for(int i=1;i<=h;i++){
    rep(j,w){
      long l,r;
      if(j == 0) l = 0;
      else{
        l = dp.at(i-1).at(j-1) * data[j-1] * data[w-1-j];
      }
      if(j == w-1) r = 0;
      else{
        r = dp.at(i-1).at(j+1) * data[j] * data[w-2-j];
      }
      long m = dp.at(i-1).at(j) * data[j] * data[w-1-j];

      dp.at(i).at(j) = (l + r + m)%MOD;
    }
  }
  cout << dp.at(h).at(k-1) % MOD << endl;
}