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: Accepted)

import Control.Monad
import Data.List
import qualified Data.IntMap.Strict as IntMap

main :: IO ()
main = do
  (n:m:_) <- fmap (map read . words) getLine :: IO [Int]
  am <- replicateM m $ map read . words <$> getLine :: IO [[Int]]
  print $ solve n am

solve :: Int -> [[Int]] -> Int
solve n am = length . filter (\as -> isValid (toAdjList am) (1:as)) $ candidates n

type AdjList = IntMap.IntMap [Int]
toAdjList :: [[Int]] -> AdjList
toAdjList am = go am IntMap.empty
  where
    go :: [[Int]] -> AdjList -> AdjList
    go [] adjList = adjList
    go (s:ab) adjList = go ab $ if a == b then new else new2
      where
        (a:b:_) = s
        new = IntMap.insertWith (++) a [b] adjList
        new2 = IntMap.insertWith (++) b [a] new

candidates :: Int -> [[Int]]
candidates n = permutations [2..n]

isValid :: AdjList -> [Int] -> Bool
isValid adjList as = goAllNodes && validateEdge adjList as
  where
    goAllNodes = all (\i -> i `elem` as) [1..(length as)]
    validateEdge :: AdjList -> [Int] -> Bool
    validateEdge adjList (a:[]) = True
    validateEdge adjList (a:b:as) = b `elem` (adjList IntMap.! a) && validateEdge adjList (b:as)