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 Ruby (2.3.3) (judge verdict: Accepted)

# Adjacency matrix graph
class AMGraph
  attr :graph, :visited

  # n: 頂点の数
  def initialize(n)
    @n = n
    reset
  end

  # like: 1, 2
  def set_side(side)
    a, b = side
    @graph[a][b] = 1
    @graph[b][a] = 1
  end

  def visit(node)
    @visited[node] = true
  end

  def recursively_visit(node)
    return if visited?(node)
    visit(node)
    visitable_nodes(node).each do |node|
      recursively_visit(node)
    end
  end

  def visited?(node)
    @visited[node]
  end

  def visitable_nodes(node)
    nodes = []
    @graph[node].each do |node|
      next if node[0] == node
      next if node[1] != 1
      next if visited?(node[0])
      nodes << node[0]
    end
    nodes
  end

  def all_visited?
    @nodes.each do |node|
      return false unless @visited[node]
    end
    true
  end

  def reset_visited
    @visited = Hash.new(false)
  end

  def reset
    @visited = Hash.new(false)
    @graph = {}
    @nodes = []
    @n.times.with_index(1) do |_,i|
      @nodes << i
      @visited[i] = false
      @graph[i] = {}
      @n.times.with_index(1) do |_,j|
        @graph[i][j] = 0
      end
    end
  end
end

n, m = gets.split.map(&:to_i)
$g = AMGraph.new(n)

m.times do
  $g.set_side(gets.split.map(&:to_i))
end

$ans = 0

def dfs(node)
  if $g.all_visited?
    $ans += 1
    return
  end
  $g.visitable_nodes(node).each do |nd|
    next if $g.visited?(nd)
    $g.visit(nd)
    dfs(nd)
    $g.visited[nd] = false
  end
end

$g.visit(1)
dfs(1)

# $g.visitable_nodes(1).each do |node|
#   $g.recursively_visit(node)
#   ans += 1 if $g.all_visited?
#   $g.reset_visited
# end

puts $ans