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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.*;
import java.util.Map;
import java.util.HashMap;

public class Main{
    static long fix = 998244353;

    public static void main(String[] args){
        Scanner sc = new Scanner(System.in);

        long ans = 1;

//        long fix = 998244353;

        int n = sc.nextInt();

        int[] d = new int[n];

        for (int i=0; i<n; i++) d[i] = sc.nextInt();

        Arrays.sort(d);

        if (d[0]!=0){
            System.out.println(0);
            return;
        }

        Map<Integer, Integer> map = new HashMap<>();

        for (int i=0; i<n; i++){
            if(map.containsKey(d[i])){
                int v = map.get(d[i]);
                v++;
                map.put(d[i], v);
            }else {
                map.put(d[i], 1);
            }
        }

//        System.out.println(map);

        List<Integer> l = new ArrayList<>();

        for (Integer val: map.values()){
            l.add(val);
        }

        int tmp = -1;
        for (Integer val: map.keySet()){
            if (val!=tmp+1){
                System.out.println(0);
                return;
            }else {
                tmp++;
            }
        }


        for (int i=1; i<l.size()-1; i++){
            ans *= (((power((long)l.get(i), (long)l.get(i+1)))%fix));
            ans %= fix;
        }
        System.out.println(ans);
//        System.out.println(power(10, 9));
//        System.out.println(power(10000, 10000));
    }

    public static long power(long x, long n){
        long sum = 1;
        while (n>0){
            if ((n&1)==1){
                sum *= x;
                sum %= fix;
            }
            x *= x;
            x %= fix;
            n>>=1;
        }
//        return sum;
        return sum;
    }

}