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 PHP (5.6.30) (judge verdict: Runtime Error)

<?php
error_reporting(0);
$stdin = file_get_contents('php://stdin');
$line = explode("\n",$stdin);
$fi = 0;
$cnt = 0;
$list = array();
$key = new stdclass();
foreach($line as $l) {
  if (strlen($l)==0) continue;
  if ($fi == 0) {
    $a = explode(" ",$l);
    $key->A = $a;
    $fi++;
    continue;
  }
  if ($fi > 0) {
    $a = explode(" ",$l);
    $key->X[] = $a;
  }
}
//N頂点
$N = $key->A[0];
//M辺
$M = $key->A[1];
//https://book.mynavi.jp/manatee/detail/id=65960
//頂点フラグ
$used = new SplFixedArray($N);
//接続状態を保存する
$connected = new SplFixedArray($N);
//初期化
for($i=0;$i<$N;$i++) {
  $used[$i]=0;
  //接続状態はconnected[頂点数=N][接続状態数=M]
  $connected[$i]=new SplFixedArray($M);
  for($j=0;$j<$M;$j++) {
    $connected[$i][$j]=0;
  }
}
//つながっているところを1にする
//頂点1-2がつながっているなら
// connected[0][1]=1
// connected[1][2]=1になるようにする
for($i=0;$i<$N;$i++) {
  $a = $key->X[$i][0]-1;
  $b = $key->X[$i][1]-1;
  $connected[$a][$b] = 1;
  $connected[$b][$a] = 1;
}
printf("%d\n",dfs(0,1));

function dfs($now,$depth) {
  global $used,$connected,$M,$N;
  if ($used[$now]) return 0;
  if ($depth == $M) return 1;
  $used[$now]=1;
  $ans  = 0;
  for($i=0;$i<$M;$i++) {
    if ($connected[$now][$i]) {
      $ans += dfs($i,$depth+1);
    }
  }
  $used[$now]=0;
  return $ans;
}
^C
<fsrv1: 230> php -f x.php
7 7
1 3
2 7
3 4
4 5
4 6
5 6
6 7
1D
<fsrv1: 231> !cat
cat x.php
<?php
error_reporting(0);
$stdin = file_get_contents('php://stdin');
$line = explode("\n",$stdin);
$fi = 0;
$cnt = 0;
$list = array();
$key = new stdclass();
foreach($line as $l) {
  if (strlen($l)==0) continue;
  if ($fi == 0) {
    $a = explode(" ",$l);
    $key->A = $a;
    $fi++;
    continue;
  }
  if ($fi > 0) {
    $a = explode(" ",$l);
    $key->X[] = $a;
  }
}
//N頂点
$N = $key->A[0];
//M辺
$M = $key->A[1];
//https://book.mynavi.jp/manatee/detail/id=65960
//頂点フラグ
$used = new SplFixedArray($N);
//接続状態を保存する
$connected = new SplFixedArray($N);
//初期化
for($i=0;$i<$N;$i++) {
  $used[$i]=0;
  //接続状態はconnected[頂点数=N][頂点数=N]
  $connected[$i]=new SplFixedArray($N);
  for($j=0;$j<$N;$j++) {
    $connected[$i][$j]=0;
  }
}
//つながっているところを1にする
//頂点1-2がつながっているなら
// connected[0][1]=1
// connected[1][2]=1になるようにする
for($i=0;$i<$M;$i++) {
  $a = $key->X[$i][0]-1;
  $b = $key->X[$i][1]-1;
  $connected[$a][$b] = 1;
  $connected[$b][$a] = 1;
}
printf("%d\n",dfs(0,1));

function dfs($now,$depth) {
  global $used,$connected,$M,$N;
  if ($used[$now]) return 0;
  if ($depth == $N) return 1;
  $used[$now]=1;
  $ans  = 0;
  for($i=0;$i<$N;$i++) {
    if ($connected[$now][$i]) {
      $ans += dfs($i,$depth+1);
    }
  }
  $used[$now]=0;
  return $ans;
}