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 Julia (0.5.0) (judge verdict: Accepted)

parseInt(x) = parse(Int, x)
parseMap(x::Array{SubString{String},1}) = map(parseInt, x)

function pos(i,j,w)
	if w == 1
		1
	elseif j <= 2 || j >= w+1
    	if i == j
      		fib(w)
    	else
      		fib(w-1)
    	end
    elseif i <= 2 || i >= w+1
		fib(w-1)
	elseif i == j
		fib(i-1)*fib(w+2-i)
	else
		fib(min(i,j)-1)*fib(w+2-max(i,j))
	end
end

function fib(k)
	if k == 1
		1
	elseif k == 2
		1
	else
		x = Array{Int}(k)
		x[1] = 1
		x[2] = 1
		for i in 3:k
			x[i] = x[i-2]+x[i-1]
		end
		x[k]
	end
end

function main()
	h,w,k = readline() |> split |> parseMap
	dp = Array{Int}(h+1,w+2)
	count = 0
	for i in 1:w+2
		dp[1,i] = 0
	end
	for i in 1:h+1
		dp[i,1] = 0
		dp[i,w+2] = 0
	end
    dp[1,2] = 1
	for i in 2:h+1
		for j in 2:w+1
			dp[i,j] = ((dp[i-1,j-1]*pos(j-1,j,w))%1000000007 + (dp[i-1,j]*pos(j,j,w))%1000000007 + (dp[i-1,j+1]*pos(j+1,j,w))%1000000007)%1000000007
		end
	end
	println(dp[h+1,k+1]%1000000007)
end

main()