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 Common Lisp (SBCL 1.1.14) (judge verdict: Wrong Answer)

(defun dfs (graph v n visited)
  (let ((all-visited-p t))
    (dotimes (i n)
      (if (null (aref visited i))
          (setf all-visited-p nil)))
    (if all-visited-p
        1
        (let ((res 0))
          (dotimes (i n)
            (if (and (aref graph v i)
                     (null (aref visited i)))
                (setf (aref visited i) t
                      res (+ res (dfs graph i n visited))
                      (aref visited i) nil)))
          res))))

(let* ((n (read))
       (m (read))
       (graph (make-array `(,n ,n) :initial-element nil))
       (visited (make-array n :initial-element nil)))
  (declare (ignore m))
  (dotimes (i n)
    (let ((a (1- (read)))
          (b (1- (read))))
      (setf (aref graph a b) t
            (aref graph b a) t)))
  (setf (aref visited 0) t)
  (format t "~A~%" (dfs graph 0 n visited)))