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 LuaJIT (2.0.4) (judge verdict: Accepted)

local mod = 1000000007
local mfl = math.floor
local h, w, K = io.read("*n", "*n", "*n")

local linepat = 1
for i = 1, w - 1 do
  linepat = linepat * 2
end
local linetbl = {}
for i = 0, linepat - 1 do
  local lineidx, mul = 0, 1
  local box = {}
  for j = 1, w - 1 do box[j] = false end
  for j = 1, w - 1 do
    if i % 2 == 1 then
      box[j] = true
      lineidx = lineidx + mul
      i = i - 1
    end
    i = i / 2
    mul = mul * 2
  end
  local isok = true
  for j = 1, w - 2 do
    if box[j] and box[j + 1] then isok = false break end
  end
  if isok then table.insert(linetbl, lineidx) end
end
-- for i = 1, #linetbl do print(linetbl[i]) end

local t, u = {}, {}
for i = 1, w do t[i], u[i] = 0, 0 end
t[1] = 1
for i = 1, h do
  for j = 1, w do u[j] = t[j] end
  for j = 2, #linetbl do
    local line_i = linetbl[j]
    local k = 1
    while k <= w - 1 do
      if line_i % 2 == 1 then
        u[k] = (u[k] + t[k + 1]) % mod
        u[k + 1] = (u[k + 1] + t[k]) % mod
        line_i = mfl((line_i - 1) / 4)
        k = k + 2
        if k == w then u[w] = (u[w] + t[w]) % mod end
      else
        u[k] = (u[k] + t[k]) % mod
        if k == w - 1 then u[k + 1] = (u[k + 1] + t[k + 1]) % mod end
        k = k + 1
        line_i = line_i / 2
      end
    end
  end
  for j = 1, w do t[j] = u[j] end
end
print(t[K])