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>
#include<stdlib.h>
#include<math.h>

typedef long long int int64;

#define MAX(a,b) ((a)>(b)?(a):(b))
#define MIN(a,b) ((a)<(b)?(a):(b))
#define ABS(a) ((a)>(0)?(a):-(a))

const int mod=1000000007;

void calc(int index,int *edge,int w,int *now,int *next){
  if(index>=w-1){
    int i;
    for(i=0;i<w;i++){
      if(i+1<w && edge[i]){
	next[i]+=now[i+1];
      } else if(i-1>=0 && edge[i-1]){
	next[i]+=now[i-1];
      } else {
	next[i]+=now[i];
      }
      next[i]%=mod;
    }
    return;
  }
  calc(index+1,edge,w,now,next);
  edge[index]=1;
  calc(index+2,edge,w,now,next);
  edge[index]=0;
  return;
}

void run(void){
  int h,w,k;
  scanf("%d%d%d",&h,&w,&k);
  int *dp=(int *)calloc(2*w,sizeof(int));
  int now=0;
  dp[0]=1;
  for(int i=1;i<=h;i++){
    int next=now^w;
    for(int j=0;j<w;j++) dp[next+j]=0;
    int edge[8]={0,0,0,0,0,0,0,0};
    calc(0,edge,w,dp+now,dp+next);
    now=next;
  }
  printf("%d\n",dp[now+k-1]);
}

int main(void){
  run();
  return 0;
}