Problem:
Score : 300 points

Problem Statement

You are given an undirected unweighted graph with N vertices and M edges that contains neither self-loops nor double edges.

Here, a self-loop is an edge where a_i = b_i (1≤i≤M), and double edges are two edges where (a_i,b_i)=(a_j,b_j) or (a_i,b_i)=(b_j,a_j) (1≤i<j≤M).

How many different paths start from vertex 1 and visit all the vertices exactly once?

Here, the endpoints of a path are considered visited.

For example, let us assume that the following undirected graph shown in Figure 1 is given.

Figure 1: an example of an undirected graph

The following path shown in Figure 2 satisfies the condition.

Figure 2: an example of a path that satisfies the condition

However, the following path shown in Figure 3 does not satisfy the condition, because it does not visit all the vertices.

Figure 3: an example of a path that does not satisfy the condition

Neither the following path shown in Figure 4, because it does not start from vertex 1.

Figure 4: another example of a path that does not satisfy the condition

Constraints

2≦N≦8

0≦M≦N(N-1)/2

1≦a_i<b_i≦N

The given graph contains neither self-loops nor double edges.

Input

The input is given from Standard Input in the following format:

N M
a_1 b_1
a_2 b_2
:
a_M b_M

Output

Print the number of the different paths that start from vertex 1 and visit all the vertices exactly once.

Sample Input 1

3 3
1 2
1 3
2 3

Sample Output 1

2

The given graph is shown in the following figure:

The following two paths satisfy the condition:

Sample Input 2

7 7
1 3
2 7
3 4
4 5
4 6
5 6
6 7

Sample Output 2

1

This test case is the same as the one described in the problem statement.

Solution in LuaJIT (2.0.4) (judge verdict: Accepted)

function dp(point, visited, vertexes, memo)
  if not string.find(visited, "0") then
    return 1
  elseif memo[point][visited] then
    return memo[point][visited]
  else
    local ans = 0
    for i = 1, #vertexes[point] do
      if string.sub(visited, vertexes[point][i], vertexes[point][i]) == "0" then
        ans = ans + dp(vertexes[point][i], string.sub(visited, 1, vertexes[point][i] - 1) .. "1" .. string.sub(visited, vertexes[point][i] + 1, #visited), vertexes, memo)
      end
    end
    memo[point][visited] = ans
    return ans
  end
end

function main()
  local input = string.gmatch(io.read(), "[^ ]+")
  local n = tonumber(input())
  local m = tonumber(input())

  local vertexes = {}
  local memo = {}
  for i = 1, n do
    table.insert(vertexes, {})
    table.insert(memo, {})
  end

  for i = 1, m do
    input = string.gmatch(io.read(), "[^ ]+")
    local a = tonumber(input())
    local b = tonumber(input())
    table.insert(vertexes[a], b)
    table.insert(vertexes[b], a)
  end

  local visited = "1"
  for i = 2, n do
    visited = visited .. "0"
  end

  print(dp(1, visited, vertexes, memo))
end

main()