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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;

class Program
{
    static string InputPattern = "InputX";

    static List<string> GetInputList()
    {
        var WillReturn = new List<string>();

        if (InputPattern == "Input1") {
            WillReturn.Add("3 3");
            WillReturn.Add("1 2");
            WillReturn.Add("1 3");
            WillReturn.Add("2 3");
            //2
        }
        else if (InputPattern == "Input2") {
            WillReturn.Add("7 7");
            WillReturn.Add("1 3");
            WillReturn.Add("2 7");
            WillReturn.Add("3 4");
            WillReturn.Add("4 5");
            WillReturn.Add("4 6");
            WillReturn.Add("5 6");
            WillReturn.Add("6 7");
            //1
        }
        else {
            string wkStr;
            while ((wkStr = Console.ReadLine()) != null) WillReturn.Add(wkStr);
        }
        return WillReturn;
    }

    struct JyoutaiDef
    {
        internal int CurrNode;
        internal HashSet<int> VisitedSet;
        //internal string Log;
    }

    static void Main()
    {
        List<string> InputList = GetInputList();

        int[] wkArr = { };
        Action<string> SplitAct = pStr =>
            wkArr = pStr.Split(' ').Select(X => int.Parse(X)).ToArray();

        SplitAct(InputList[0]);
        int N = wkArr[0];

        bool[,] EdgeArr = new bool[N + 1, N + 1];

        foreach (string EachStr in InputList.Skip(1)) {
            SplitAct(EachStr);
            EdgeArr[wkArr[0], wkArr[1]] = true;
        }

        var Stk = new Stack<JyoutaiDef>();
        JyoutaiDef WillPush;
        WillPush.CurrNode = 1;
        WillPush.VisitedSet = new HashSet<int>() { 1 };
        //WillPush.Log = "1";
        Stk.Push(WillPush);

        int AnswerCnt = 0;
        while (Stk.Count > 0) {
            JyoutaiDef Popped = Stk.Pop();

            //クリア判定
            if (Popped.VisitedSet.Count == N) {
                AnswerCnt++;
                //Console.WriteLine(Popped.Log);
                continue;
            }

            for (int I = 1; I <= N; I++) {
                //再訪不可
                if (Popped.VisitedSet.Contains(I)) continue;

                Action<int> PushSyori = (pNewNode) =>
                {
                    WillPush.CurrNode = pNewNode;
                    WillPush.VisitedSet = new HashSet<int>(Popped.VisitedSet);
                    WillPush.VisitedSet.Add(pNewNode);
                    //WillPush.Log = Popped.Log + "," + pNewNode.ToString();

                    Stk.Push(WillPush);
                };

                if (EdgeArr[Popped.CurrNode, I]) {
                    PushSyori(I);
                }
                if (EdgeArr[I, Popped.CurrNode]) {
                    PushSyori(I);
                }
            }
        }
        Console.WriteLine(AnswerCnt);
    }
}