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){
    Scanner sc = new Scanner(System.in);
    int n = sc.nextInt();
	int[] d = new int[n];
    d[0] = sc.nextInt();
    if(d[0]!=0){
      System.out.println(0);
    }
    else{
      for(int i=1; i<n; i++){
        d[i] = sc.nextInt();
    	}
      Arrays.sort(d);
      System.out.println(Check(d));
  	}
  }
  
  public static int Check(int[] d){
    int ans = 1;
    int cnt = 1;
    for(int i=0; i<d.length-1; i++){
      if(d[i]==d[i+1] && d[i+1]!=0){
        cnt++;
      }
      else if(d[i]==d[i+1]-1 && d[i+1] != 0){
        ans = ans*Factorial(cnt)*Factorial(ans);
        cnt = 1;
      }
      else{
        ans = 0;
      }
    }
    return ans%998244353;
  }
  
  public static int Factorial(int x){
    if(x == 1){
      return 1;
    }
    else{
      return x * Factorial(x-1);
    } 
  }
}