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 number_of_amidakuji
  implicit none
  integer(8), parameter :: md = 1000000007_8
  integer :: h, w, k, i, j
  integer(8) :: dp(0:100,0:9) = 0_8, f(-1:9) = 0_8
  dp(0,1) = 1_8
  f(0) = 1_8
  read(*,*) h, w, k
  do i = 1, w
    f(i) = mod(f(i-1)+f(i-2),md)
  end do
  do i = 1, h
    do j = 1, w
      dp(i,j) = mod(mod(f(j-2)*f(w-j)*dp(i-1,j-1),md) &
      & +mod(f(j-1)*f(w-j)*dp(i-1,j),md)+mod(f(j-1)*f(w-j-1)*dp(i-1,j+1),md),md)
    end do
  end do
  write(*,'(i0)') dp(h,k)
end program number_of_amidakuji