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 Kotlin (1.3.71) (judge verdict: Accepted)

fun main() {
  val (N, M) = readInts()
  val edges = Array(N + 1) { mutableSetOf<Int>() }
  repeat(M) {
    val (a, b) = readInts()
    edges[a].add(b)
    edges[b].add(a)
  }
  fun isOk(list: List<Int>): Boolean {
    if (list.first() != 1) return false
    for (i in 0 until N - 1) {
      val a = list[i]
      val b = list[i + 1]
      if (!edges[a].contains(b)) return false
    }
    return true
  }

  val permList = permute((1..N).map { it })
  var sum = 0L
  for (perm in permList) {
    if (isOk(perm)) sum++
  }
  println(sum)
}

fun <Int> permute(list: List<Int>): List<List<Int>> {
  if (list.size == 1) return listOf(list)
  val perms = mutableListOf<List<Int>>()
  val sub = list[0]
  for (perm in permute(list.drop(1)))
    for (i in 0..perm.size) {
      val newPerm = perm.toMutableList()
      newPerm.add(i, sub)
      perms.add(newPerm)
    }
  return perms
}

fun readString() = readLine()!!
fun readStrings() = readString().split(" ")
fun readInt() = readString().toInt()
fun readLong() = readString().toLong()
fun readInts() = readStrings().map { it.toInt() }
fun readLongs() = readStrings().map { it.toLong() }