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 sc = new Scanner(System.in);
    int n = sc.nextInt();
    int[] count = new int[n + 1];
    int max = 0;
    for(int i = 0; i < n; i++) {
      int num = sc.nextInt();
      count[num]++;
      max = Math.max(max, num);
    }
    if (count[0] != 1) {
      System.out.println(0);
      return;
    }
    for(int i = max; i >= 0; i--) {
      if (count[i] == 0) {
        System.out.println(0);
        return;
      }
    }
    long ans = 1;
    for(int i = 1; i <= max; i++) {
      long prev = count[i - 1];
      long base = 1;
      long num = count[i];
      for(int j = 0; j < num; j++) {
        base *= prev;
        base %= 998244353;
      }
      ans *= base;
      ans %= 998244353;
    }
    System.out.println(ans);
  }
}