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.*;
public class Main {
    public static void main(String args []){
        Scanner sc = new Scanner(System.in);
        int N = sc.nextInt();
        int [] List = new int[N+1];
        long opt=1;
        int root = 1;
        int max = 0;
        for(int i=0;i<N;i++) {
            int num = sc.nextInt();
            if(i==0) root = num;
            if(max<num) max = num;
            List[num]++;
        }
        if(root==0&&List[0]==1){
            for(int i=1;i<max;i++){
                if(List[i]!=0&&List[i+1]!=0){
                    long num2 = 1;
                    for(int j=0;j<List[i+1];j++)num2= (num2*List[i])%998244353;
                    opt=(opt*num2)%998244353;
                }
                else{
                    opt = 0;
                    break;
                }
            }

        }
        else  opt=0;
        System.out.println(opt);
    }
}