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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace ConsoleApp1
{
    class Program
    {
        static void Main(string[] args)
        {
            int n = int.Parse(Console.ReadLine());
            String[] d_str = Console.ReadLine().Split();

            List<int> d = new List<int>();
            for (int i = 0; i < n; i++) {
                d.Add(int.Parse(d_str[i]));
            }

            List<int> num = new List<int>();
            for (int i = 0; i < n; i++) {
                num.Add(0);
            }

            for (int i = 0; i < n ; i++) {
                num[d[i]]++;
            }

            double ans = 0;
            if (d[0] != 0) {
                ans = 0;
            } else if (num[0] != 1) {
                ans = 0;
            } else {
                ans = 1;
                for (int i = 0; i < n - 1; i++) {
                    for (int j = 0; j < num[i + 1]; j++) {
                        ans *= num[i] % 998244353;
                      ans %= 998244353;
                    }
                }
            }

            Console.WriteLine(ans % 998244353);

        }
    }

}