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 Haskell (GHC 7.10.3) (judge verdict: Runtime Error)

import Data.List
import Control.Monad

main = do
  [n,m] <- map read . words <$> getLine :: IO [Int]
  xs <- replicateM n (map read . words <$> getLine) :: IO [[Int]]
  let ps = paths n xs
  print $ trav m 1 ps []

paths :: Int -> [[Int]] -> [(Int,[Int])]
paths 0 xs = []
paths n xs = 
  let 
    f = filter (\[a,_] -> a==n) xs
  in
    if null f then (paths (n-1) xs) else (n, last $ transpose f) : (paths (n-1) xs)

trav :: Int -> Int -> [(Int,[Int])] ->  [Int] -> Int
trav m current paths arrived
  | length arrived == m-1 = 1 
  | current `elem` arrived = 0
  | otherwise = 
    let 
      ds = filter (\(a,_) -> a==current) paths
    in
      if null ds 
         then if length (current:arrived) == m-1 then 1 else 0
         else sum $ [r | d<-(snd $ head ds), let r = trav m d paths (current:arrived)]