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 Haskell (GHC 7.10.3) (judge verdict: Accepted)

fib (-1) = 0
fib 0 = 1
fib 1 = 1
fib 2 = 2
fib 3 = 3
fib 4 = 5
fib 5 = 8
fib 6 = 13
fib 7 = 21
fib _ = 0
-- fib n = (fib (n - 1)) + (fib (n - 2))
-- fib n = [1, 1, 2, 3, 5, 8, 13, 21] !! n

solve h w k =
    let
        solve' 0 m = m
        solve' h m =
            solve' (h - 1)
            ( [0] ++ [
                sum
                    [ (m !! i) * (fib (i - 1)) * (fib (w - i))
                    , (m !! (i - 1)) * (fib (i - 2)) * (fib (w - i))
                    , (m !! (i + 1)) * (fib (i - 1)) * (fib (w - i - 1))
                    ] `mod` (10 ^ 9 + 7 :: Int)
            | i <- [1..w] ] ++ [0] )
    in
        (solve' h [ if x == k then 1 else 0 | x <- [0..w+1] ]) !! 1

main = do
    [h, w, k] <- map read . words <$> getLine :: IO [Int]
    print $ solve h w k