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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
class Program
    {
        static Dictionary<int, List<int>> Nodes;
        static void Main(string[] args)
        {
            //C - One-stroke Path
            //https://atcoder.jp/contests/abc054/tasks/abc054_c
            var Array = Console.ReadLine().Split().Select(x => int.Parse(x)).ToArray();
            int N = Array[0];
            int M = Array[1];
            Nodes = new Dictionary<int, List<int>>();
            for (int i = 1; i <= N; i++)
            {
                Nodes.Add(i, new List<int>());
            }
            int Count = 0;
            while (Count < M)
            {
                Array = Console.ReadLine().Split().Select(x => int.Parse(x)).ToArray();
                //無向グラフなのでどちらからでもいける
                Nodes[Array[0]].Add(Array[1]);
                Nodes[Array[1]].Add(Array[0]);
                Count++;
            }
            int Result = CheckPath(1, new List<int>());
        }

        static int CheckPath(int CurrentNodeID, List<int> CheckedNum)
        {
            CheckedNum.Add(CurrentNodeID);
            //まだ調べていないものが次のNodeとなる
            var Next = Nodes[CurrentNodeID].Where(x => !CheckedNum.Exists(y => x == y));
            
            //次のNodeがなく、調べたノード数と全ノード数一致
            if (Next.Count() == 0 && (CheckedNum.Count == Nodes.Count()))
            {
                return 1;
            }
            //再起呼び出し
            var X = Next.Select(z => CheckPath(z, new List<int>(CheckedNum)));
            var Y = X.Sum();
            return Y;
        }
    }