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# (.NET Core 3.1.201) (judge verdict: Accepted)

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


class Program
{
    static void Main(string[] args)
    {
        const long mod = 998244353;

        // 入力
        long n = long.Parse(Console.ReadLine());
        long[] d = Console.ReadLine().Split().Select(long.Parse).ToArray();


        var max = d.Max();
        var array = new long[max + 1];
        for (int i = 0; i < d.Length; i++)
        {
            array[d[i]]++;
        }

        long ans = 1;
        long count = 0;
        long parent = 0;

        if (d[0] != 0 || array[0] != 1)
        {
            ans = 0;
        }
        else
        {
            for (int i = 1; i < array.Length; i++)
            {
                for (int j = 0; j < array[i]; j++)
                {
                    ans *= array[i - 1];
                    ans %= mod;
                }
            }
        }

        // 出力
        Console.WriteLine(ans);
    }
}