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: Runtime Error)

import java.util.*;

class Main {
    public static void main(String[] args) {
        //final int MOD = 998244353;
        Scanner sc = new Scanner(System.in);
        int N = Integer.parseInt(sc.next());
        int D[] = new int[N];
        for (int i=0; i<N; i++)
            D[i] = Integer.parseInt(sc.next());
        sc.close();

        Map<Integer, Integer> map = new HashMap<>();
        for (int i=0; i<N; i++) {
            if (map.containsKey(D[i])) {
                map.put(D[i], map.get(D[i])+1);
            } else {
                map.put(D[i], 1);
            } 
        }

        long cnt = 1;
        for (int i=0; i<map.size(); i++) {
            if (i>0) {
                if (!map.containsKey(i)) {
                    cnt = 0;
                } else {
                    cnt *= mPow(map.get(i-1) ,map.get(i));
                }
            } else if (!map.containsKey(0) || map.get(0)!=1) {
                cnt = 0;
                break;
            }
        }
        System.out.println(cnt);
    }
    public static long mPow(int a, int b) {
        if (b==1) {
            return a;
        } else {
            return a * mPow(a, b-1) % 998244353L;
        }
    }
}