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 Scala (2.11.7) (judge verdict: Accepted)

import java.util.Scanner

object Main {
  def solve(N: Int, e: Array[Array[Boolean]]): Int = {
    permutation((2 to N).toArray, 0){ (p, acc) =>
      val path = 1 +: p
      val legit = path zip path.drop(1) forall { case (i, j) =>
        e(i)(j)
      }
      acc + (if(legit) 1 else 0)
    }
  }

  def main(args: Array[String]): Unit = {
    val sc = new Scanner(System.in)
    val N = sc.nextInt()
    val M = sc.nextInt()
    val e = Array.ofDim[Boolean](N + 1, N + 1)
    (0 until M) foreach { _ =>
      val a = sc.nextInt()
      val b = sc.nextInt()
      e(a)(b) = true
      e(b)(a) = true
    }
    println(solve(N, e))
  }

  @specialized
  def permutation[A, B](a: Array[A], z: B)(f: (Array[A], B) => B): B = {
    def swap(i: Int, j: Int): Unit = {
      val temp = a(i)
      a(i) = a(j)
      a(j) = temp
    }

    def step(n: Int, acc: B): B = n match {
      case 1 => f(a, acc)
      case _ =>
        val acc2 = (0 until n - 1).foldLeft(acc) { (acc0, i) =>
          val acc1 = step(n - 1, acc0)
          if (n % 2 == 0) {
            swap(i, n - 1)
          } else {
            swap(0, n - 1)
          }
          acc1
        }
        step(n - 1, acc2)
    }

    step(a.length, z)
  }
}