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.io.BufferedReader
import java.util.Arrays

fun main(args: Array<String>) {
    /** comment in when you submit **/
    System.`in`.bufferedReader().use { println(getOutputString(it)) }

    /** delete below after your test **/
    // executeTest()
}

private fun getOutputString(reader: BufferedReader): String {
    val num = reader.readLine().split(" ")[0].toInt()

    var pairs = reader.readLines().map {
        val ints = it.split(" ").map { it.toInt() }
        return@map "${ints[0]}${ints[1]}"
    }
    val origPairs = pairs

    while (pairs.size > 0 && pairs.first().length < num) {
        val newPairs = mutableListOf<String>()
        origPairs.forEach { p1 ->
            pairs.forEach { p2 ->
                val candidate = when {
                    p1.first() == p2.last() -> p2 + p1.substring(1)
                    p1.first() == p2.first() -> p2 + p1.reversed().substring(1)
                    p1.last() == p2.last() -> p1 + p2.reversed().substring(1)
                    p1.last() == p2.first() -> p1 + p2.substring(1)
                    else -> null
                }

                candidate?.let {
                    val chars = it.toCharArray()
                    Arrays.sort(chars)
                    for (i in 0 until chars.size - 1) {
                        if (chars[i] == chars[i + 1]) return@let
                    }

                    newPairs.add(it)
                }
            }
        }

        pairs = newPairs.distinct().sortedBy { it.length }
    }

    return pairs.filter { it.startsWith("1") && it.length == num }.size.toString()
}