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)

module Main where
import Control.Monad
import Control.Applicative
import Data.Maybe
import Data.List
import qualified Text.Printf
import qualified Data.ByteString.Char8 as BC
import qualified Data.ByteString as BC

------------------------------------------

main :: IO ()
main = do
  [n, m] <- readInts
  edges <- replicateM m $ do
    [a, b] <- readInts
    return (a-1, b-1)
  let graph = createGraph n edges
  print $ solve n graph

type Graph = [[Int]]

createGraph :: Int -> [(Int, Int)] -> Graph
createGraph n edges = loop (replicate n []) edges
  where updateGraph i v graph = concat [take i graph, [v : (graph !! i)], drop (i+1) graph]
        loop graph [] = graph
        loop graph ((a,b):es) =
          let nextGraph = updateGraph b a $ updateGraph a b graph
          in loop nextGraph es

solve :: Int -> Graph -> Int
solve n graph = dfs 0 1 [0]
  where dfs i depth visited
          | depth == n = 1
        dfs i depth visited
          | nexts == [] = 0
          | otherwise = sum $ flip map nexts $ \w -> dfs w (depth+1) (w:visited)
            where nexts = filter (not . (`elem` visited)) $ graph !! i
          -- in sum $ flip map nexts $ \w -> dfs w (depth+1) (w:visited)

------------------------------------------

{- Int input -}

parseInt :: BC.ByteString -> Int
parseInt = fst . fromJust . BC.readInt

parseInts :: BC.ByteString -> [Int]
parseInts = map parseInt <$> BC.words

readInt :: IO Int
readInt = parseInt <$> BC.getLine

readInts :: IO [Int]
readInts = parseInts <$> BC.getLine

{- Double Formatting -}

doubleFmt :: Double -> String
doubleFmt = Text.Printf.printf "%.12f"