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 <math.h>
#define Lim 1000000007

int main( void ){
	int H,W,K;
	int pat[63][10]={0};
	int i,j,b,flg,cnt=0;
	unsigned long count[10]={0,1,0,0,0,0,0,0,0,0},count2[10];
	
	scanf("%d%d%d",&H,&W,&K);
	for(i=0;i<pow(2,W-1);i++){
		b=i;
		flg=1;
		for(j=1;j<9;j++){
			pat[cnt][j]=b%2;
			b=b/2;
			if( pat[cnt][j]==1 && pat[cnt][j-1]==1 ){
				flg=0;
				j=9;
			}
		}
		if( flg==1 ) cnt++;
	}

	for(i=0;i<H;i++){
		for(j=0;j<10;j++) count2[j]=0;
		for(j=0;j<cnt;j++){
			for(b=1;b<9;b++){
				if( pat[j][b]==1 ) count2[b+1]+=count[b];
				else if( pat[j][b-1]==1 ) count2[b-1]+=count[b];
				else count2[b]+=count[b];
				count2[b]=count2[b]%Lim;
			}
		}
		for(j=0;j<10;j++) count[j]=count2[j];
	}
	printf("%ld\n",count[K]%Lim);
	return 0;
}