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 (1.0.0) (judge verdict: Accepted)

import java.util.*

fun main(args: Array<String>) {
    val sc = Scanner(System.`in`)
    val n = sc.next().toInt()
    val m = sc.next().toInt()
    val a = mutableListOf<Int>()
    val b = mutableListOf<Int>()
    repeat((0 until m).count()) {
        a.add(sc.next().toInt())
        b.add(sc.next().toInt())
    }
    println(problem54c(n, m, a, b))
}

fun problem54c(n: Int, m: Int, a: List<Int>, b: List<Int>): Int {
    var count = 0
    fun dfs(v: Int, past: HashSet<Int>) {
//        debugLog("v", v, "past", past)
//        debugLog("if", past.sorted(), (a + b).toHashSet().sorted())
        if (past.sorted() == (a + b).toHashSet().sorted()) {
//            debugLog("success")
            count++
            return
        }
        val nextList = mutableListOf<Int>()
        (0 until m).forEach { i ->
            val ai = a[i]
            if (!past.contains(ai) && v == b[i]) {
                nextList.add(ai)
            }
            val bi = b[i]
            if (!past.contains(bi) && v == a[i]) {
                nextList.add(bi)
            }
        }
//        debugLog("nextList", nextList)
        for(next in nextList) {
            if (past.contains(next)) continue
            past.add(next)
            dfs(next, past.toHashSet())
            past.remove(next)
        }
    }
    dfs(1, hashSetOf(1))
    return count
}