Problem:
Score : 300 points

Problem Statement

Given is an integer sequence D_1,...,D_N of N elements. Find the number, modulo 998244353, of trees with N vertices numbered 1 to N that satisfy the following condition:

For every integer i from 1 to N, the distance between Vertex 1 and Vertex i is D_i.

Notes

A tree of N vertices is a connected undirected graph with N vertices and N-1 edges, and the distance between two vertices are the number of edges in the shortest path between them.

Two trees are considered different if and only if there are two vertices x and y such that there is an edge between x and y in one of those trees and not in the other.

Constraints

1 \leq N \leq 10^5

0 \leq D_i \leq N-1

Input

Input is given from Standard Input in the following format:

N
D_1 D_2 ... D_N

Output

Print the answer.

Sample Input 1

4
0 1 1 2

Sample Output 1

2

For example, a tree with edges (1,2), (1,3), and (2,4) satisfies the condition.

Sample Input 2

4
1 1 1 1

Sample Output 2

0

Sample Input 3

7
0 3 2 1 2 2 1

Sample Output 3

24

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

import Data.Char
import Data.Maybe
import Data.List
import qualified Data.ByteString.Char8 as B
import qualified Data.Vector.Unboxed as V

readInt = B.readInt . B.dropWhile isSpace

getInt = fst . fromJust . readInt <$> B.getLine

getIntList = unfoldr readInt <$> B.getLine

main = do
  n <- getInt
  ds <- map (\s -> (head s, length s)) . group . sort <$> getIntList
  let len = length ds
  let es = V.accum (+) (V.replicate len 0) ds
  if V.all (/= 0) es
    then print . V.foldr (((`mod` modInt) .) . (*)) 1 . V.zipWith (((`mod` modInt) .) . (^)) es $ V.tail es
    else print 0

modInt = 998244353