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 Lua (5.3.2) (judge verdict: Accepted)

local mod = 1000000007

local function bmul(x, y) return (x * y) % mod end
local function badd(x, y) return (x + y) % mod end

local h, w, k = io.read("*n", "*n", "*n")
if w == 1 then
  print(1)
else
  local t = {1}
  for i = 2, w do t[i] = 0 end
  local tot = 2^(w - 1)
  local map = {}
  for i = 1, w do
    map[i] = {}
    for j = 1, w do
      map[i][j] = 0
    end
  end
  for it = 0, tot - 1 do
    local valid = true
    local flag = {}
    for i = 1, w - 1 do
      flag[i] = it % 2 == 1
      it = it // 2
    end
    for i = 1, w - 2 do
      if flag[i] and flag[i + 1] then
        valid = false
      end
    end
    if valid then
      for i = 1, w do
        if i == 1 then
          if flag[1] then
            map[1][2] = map[1][2] + 1
          else
            map[1][1] = map[1][1] + 1
          end
        elseif i == w then
          if flag[w - 1] then
            map[w][w - 1] = map[w][w - 1] + 1
          else
            map[w][w] = map[w][w] + 1
          end
        else
          if flag[i] then
            map[i][i + 1] = map[i][i + 1] + 1
          elseif flag[i - 1] then
            map[i][i - 1] = map[i][i - 1] + 1
          else
            map[i][i] = map[i][i] + 1
          end
        end
      end
    end
  end
  for ih = 1, h do
    local nt = {}
    for i = 1, w do nt[i] = 0 end
    for src = 1, w do
      for dst = 1, w do
        nt[dst] = badd(nt[dst], bmul(t[src], map[src][dst]))
      end
    end
    for i = 1, w do t[i] = nt[i] end
  end
  print(t[k])
end