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.Scanner;

public class Main{
    public static void main(String[] args){
        Scanner sc = new Scanner(System.in);
        int N = sc.nextInt();
        int D[] = new int[N];
        int counter[] = new int[N];

        for(int i=0; i < N; i++) {
            D[i] = sc.nextInt();
            counter[D[i]]++;
            // System.out.println("c[" + D[i] + "]: " + counter[D[i]]);
        }

        if(D[0] != 0 || counter[0] >= 2){
            System.out.println(0);
        }else{
          long sum = 1;
          for(int i = 1; i < N; i++) {
              for(int j = 0; j < counter[i]; j++) {
                sum *= counter[i-1];
                sum %= 998244353L;
              }
          }
          System.out.println(sum);
        }
    }
}