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 Fortran (gfortran v4.8.4) (judge verdict: Accepted)

program test
implicit none

integer(8) :: h,w,k,i,j,ans,big=1d9+7
integer(8) :: go(0:10,3)=0,bou(0:10)=0,dp(0:200,0:10)=0

read(*,*) h,w,k

if(w == 1)then
	write(*,*) 1
else

	!全探査するか…
	do i=0,2**(w-1)-1
		do j=1,w-1
			if(BTEST(i,j-1))then
				bou(j) = 1
			else
				bou(j) = 0
			endif
		enddo
		
		do j=1,w-2
			if(bou(j)+bou(j+1)==2)then
				goto 100
			endif
		enddo
		
		!i番目にいるときに、左に行くか右に行くかそうでないか
		do j=1,w
			if(bou(j-1) == 1) then !左
				go(j,1) = go(j,1) + 1
			else if(bou(j) == 1) then !右
				go(j,3) = go(j,3) + 1
			else
				go(j,2) = go(j,2) + 1
			endif
		enddo
		
		100 continue
		
	enddo




	!とりあえずDPっしょ
	dp(0,1)=1
	do i=1,h
		do j=1,w
			!j-1から右へ、jからそのまま、j+1から左へ
			dp(i,j) = mod(dp(i-1,j-1)*go(j-1,3) +dp(i-1,j)*go(j,2) + dp(i-1,j+1)*go(j+1,1),big)
		enddo
	enddo

	write(*,*) dp(h,k)

endif

end program