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: Accepted)

import java.util.*;
import java.io.PrintWriter;
import static java.lang.Integer.*;
import static java.lang.Long.*;
import static java.lang.Math.*;
import static java.lang.System.*;

public class Main {
	public static long MOD = 998244353;
	public static void main(String[] args) {
		int i,j;
		Scanner sc = new Scanner(in);
		int n = parseInt(sc.next());
		int[] d = new int[n];
		HashMap<Integer, Integer> map = new HashMap<>();
		for(i=0;i<n;i++) {
			d[i] = parseInt(sc.next());
			if(map.get(d[i])==null) {
				map.put(d[i], 1);
			} else {
				map.put(d[i], map.get(d[i])+1);
			}
		}
		sc.close();
		long ans=0;
		if(d[0]==0) {
			ArrayList<Integer> list = new ArrayList<>(map.keySet());
			Collections.sort(list);
			boolean f=true;
			for (i = 1;i < list.size(); i++) {
				if(list.get(i)-list.get(i-1)!=1) {
					f=false;
					break;
				}
			}
			if(map.get(list.get(0))>1)f=false; 
			if(f) {
				ans=1;
				i=0;
				int pre = list.get(i);
				i++;
				if(pre==0) {
					pre = list.get(i);
					i++;
				}
				for (; i < list.size(); i++) {
					for (j = 0; j < map.get(list.get(i)); j++) {
						ans*=(long)map.get(pre);
						if(ans>=MOD)ans%=MOD;
					}
					pre = list.get(i);
				}
			}
		}
		out.println(ans);
	}
}