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: Wrong Answer)

import java.util.*;

public class Main {
	  public static void main(String args[]){
      	Scanner scan = new Scanner(System.in);
            
        int N = scan.nextInt();
        int D[] = new int[N];
        int num_length[] = new int[N];
        int max_length = 0;
        
        boolean bad_input = false;
        
        long cnt = 1;
        
        //標準入力-----------------------
        if( N != 1 )
        for( int i = 0 ; i < N ; i++ ){
          D[i] = scan.nextInt();
          
          if( N-1 <  D[i] ){
            //exception
            bad_input = true;
            cnt = 0;
            break;
          }
          num_length[ D[i] ] += 1;
          
          if(max_length < D[i])max_length=D[i];
        }
        //-------------------------------
        if( D[0] != 0 ){
          cnt = 0;
        }
        else if( num_length[0] != 1 ){
          cnt = 0;
        }
        else if(!bad_input){
          for( int i = 1 ; i < max_length ; i++ ){
            //powだと数字がでかい時にエラーになる
            //cnt *= (long)Math.pow( num_length[i] , num_length[i+1]  );
            for( int j = 0 ; j < num_length[i+1] ; j++){
              cnt = (long)cnt * (long)num_length[i];
            }
          }
        }
         
        //cnt = (long)998244354 * (long)3;
        //cnt = (long)Math.pow( 998244354 , 2  );
        //cnt = (long)998244354 * (long)998244354;
            
        System.out.println(cnt % (long)998244353);
	  }
}