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);
        long MOD = 998244353;
        int n = sc.nextInt();
        int node[] = new int[n];
        for(int i=0;i<n;i++)node[i]=0;
        int d = sc.nextInt();
        if(d!=0){
            System.out.println(0);
            return;
        }
        node[0]++;
        for(int i=1;i<n;i++){
            d = sc.nextInt();
            node[d]++;
        }


        if(node[0]!=1){
            System.out.println(0);
            return;
        }
        long ans = 1;
        for(int i=1;i<n;i++){
            for(int j=0;j<node[i];j++){
                ans *= node[i-1];
                ans %= MOD;
            }
        }
        System.out.println(ans);



    }
}