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)

{-# LANGUAGE BangPatterns #-}

module Main
  ( main
  )
where

import           Control.Applicative
import           Control.Monad
import           Data.Bits
import qualified Data.ByteString.Char8         as BS
import           Data.Char
import           Data.List
import           Data.Maybe
import qualified Data.Vector                   as V
import qualified Data.Vector.Mutable           as MV
import           Control.Monad.ST

parse = fst . fromJust
parseInt = parse . BS.readInt
parseInteger = parse . BS.readInteger
readInt = parseInt <$> BS.getLine
readInteger = parseInteger <$> BS.getLine
readIntList = map parseInt . BS.words <$> BS.getLine
readIntegerList = map parseInteger . BS.words <$> BS.getLine

res = 1000000000 + 7

main = do
  [n, m] <- readIntList
  con    <- concat <$> replicateM
    m
    ((\[a, b] -> [(a - 1, b - 1), (b - 1, a - 1)]) <$> readIntList)
  let matrix     = createMatrix n con
      perm       = map (0:) $ permutations [1 .. (n - 1)]
      validPaths = filter (judge matrix) perm
  print . length $ validPaths

judge matrix (a : b : rest) = ((matrix V.! a) V.! b) && judge matrix (b : rest)
judge matrix _ = True

type AdjacencyMatrix = V.Vector (V.Vector Bool)
createMatrix :: Int -> [(Int, Int)] -> AdjacencyMatrix
createMatrix n con =
  V.fromList
    $ map
        (\conByStart -> V.create $ do
          adj <- MV.replicate n False
          forM_ conByStart (\(i, j) -> MV.write adj j True)
          return adj
        )
    . groupBy (\a b -> fst a == fst b)
    . sort
    $ con