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,br_a(0:10),br_b(0:10),dp(0:200,0:10)=0

read(*,*) h,w,k

!最後に橋あり→a,なし→b

br_a(0)=0
br_b(0)=0

br_a(1)=0
br_b(1)=1

do i=2,w
	br_a(i) = br_b(i-1)
	br_b(i) = mod(br_b(i-1) + br_a(i-1),big)
enddo

!左に行く
do i=1,w
	if(i==1) then
		go(i,1) = 0
	else
		go(i,1) = mod(br_b(i-1) * br_b(w-i+1),big)
	endif
enddo

!右に行く
do i=1,w
	go(i,3) = go(w+1-i,1)
enddo

!どちらでもない
do i=1,w
	go(i,2) = mod((br_a(w)+br_b(w)) - go(i,1) - go(i,3),big)
enddo

!とりあえずDPっしょ
dp(0,1)=1
do i=1,h
	do j=1,w
		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)

end program