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>
int main(){
  int h,w,n,i,j,k,nk,M=1e9+7,d[110][10]={1};
  scanf("%d %d %d",&h,&w,&n);
  for(i=0;i<h;i++){
    for(j=0;j<(1<<(w-1));j++){
      for(k=0;k<w-1;k++){
	if((j&(1<<k))&&(j&(1<<(k+1))))goto NEXT;
      }
      //if(i==0)printf("%d\n",j);
      for(k=0;k<w;k++){
	nk=k;
	if(   j&(1<< k   ))nk=k+1;
	if(k&&j&(1<<(k-1)))nk=k-1;
	//printf("%d %d\n",nk,k);
	d[i+1][nk]=(d[i+1][nk]+d[i][k])%M;
      }
    NEXT:;
    }
  }
  /*for(i=0;i<=h;i++){
    for(j=0;j<w;j++)printf("%d ",d[i][j]);printf("\n");
    }
    printf("%d\n",n);//*/
  printf("%d\n",d[h][n-1]);
  return 0;
}