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)

local n, m = io.read("*n", "*n")
local t = {}
for i = 1, n do t[i] = {} end
for i = 1, m do
  local a, b = io.read("*n", "*n")
  t[a][b], t[b][a] = true, true
end
local mult = 1
for i = 2, n - 1 do
  mult = mult * i
end
local cnt = 0
for i = 0, mult - 1 do
  local rem = {}
  local resarray = {1}
  for j = 2, n do rem[j - 1] = j end
  local it = i

  local tmpmult = mult
  for j = n - 1, 1, -1 do 
    local divfactor = tmpmult / j
    local pos = 1 + math.floor(it / divfactor)
    table.insert(resarray, rem[pos])
    table.remove(rem, pos)
    tmpmult = divfactor
    it = it % divfactor
  end
  local isok = true
  for j = 1, n - 1 do
    if(not t[resarray[j]][resarray[j+1]]) then
      isok = false
      break
    end
  end
  if(isok) then cnt = cnt + 1 end
end
print(cnt)