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 Perl (v5.18.2) (judge verdict: Accepted)

use strict;
use warnings;

my $g = [];

sub main {
    chomp(my $s = <STDIN>);
    my ($N,$M) = split(/ /,$s);
    my (@ab);
    for(1..$M){
        chomp(my $tmp = <STDIN>);
        push @ab, [split(/ /, $tmp)];
    }
    print solver($N,$M,\@ab);
}
main() unless caller;

sub solver {
    my ($N,$M,$ab) = @_;
    my %x;

    for my $i(0..$M-1) {
        my $u = $ab->[$i]->[0];
        my $v = $ab->[$i]->[1];
        $g->[$u] = $g->[$u] ? $g->[$u] : [];
        $g->[$v] = $g->[$v] ? $g->[$v] : [];
        $g->[$u-1]->[$v-1] = 1;
        $g->[$v-1]->[$u-1] = 1;
    }
    my $visited = [];
    for my $i(0..$N-1){
        $visited->[$i]=0;
    }
    $visited->[0]=1;
    return dfs(0,$N,$visited);
}

sub dfs {
    my ($v, $N, $visited) = @_;
    
    my $all_visited = 1;
    
    for my $i(0..$N-1){
        if($visited->[$i] == 0){
            $all_visited = 0;
        }
    }
    if($all_visited){
        return 1;
    }
    my $ret=0;

    for my $i(0..$N-1){
        next if(!$g->[$v]->[$i]);
        next if($visited->[$i]);

        $visited->[$i]=1;
        $ret += dfs($i,$N,$visited);
        $visited->[$i]=0;
    }

    return $ret;
}