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 is_valid(v::Int)
    while(v > 0)
        if ((v&3) == 3) return false; end
        v ÷= 2;
    end
    return true
end
function amida(v::Int, pos::Vector{Int})
    for i = 0:length(pos)-2
        if( (v & (1 << i)) > 0)
            pos[i+1], pos[i+2] = pos[i+2], pos[i+1]
        end
    end
    return pos
end
const M = 10^9+7
g(n,A,R=A^0)=n>0 ? g(n÷2,A*A.%M,R*A^(n%2).%M) : R
function main()
    h,w,k = readline() |> split |> parsemap
    A = zeros(Int,w,w)
    for i = 0:(1 << (w-1))-1
        if (!is_valid(i)) continue end
        v = amida(i, Vector(1:w))
        for j=1:w
            @inbounds A[j,v[j]]  += 1
        end
    end
    r = g(h,A)
    println(r[1,k])
end

main()