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;

class Program
{
    static void Main()
    {
        Console.WriteLine(Solve());
        //Solve();
        return;
    }

    public static string Solve()
    {
        long N = ReadLong();
        long[] D = ReadLongs();
        long mod = 998244353;
        long ans = 1;
        long target = 1;
        long cnt = 0;

        if (D[0] != 0)
        {
            return "0";
        }
        Array.Sort(D);
        if (D[1] != 1)
        {
            return "0";
        }
        List<long> Dc = new List<long>();
        for (int i = 1; i < N; i++)
        {
            if (D[i] == target)
            {
                cnt++;
            }
            else if (D[i] == target + 1)
            {
                Dc.Add(cnt);
                target++;
                cnt = 1;
            }
            else
            {
                return "0";
            }
        }
        Dc.Add(cnt);
        for (int i = 0; i < Dc.Count - 1; i++)
        {
            ans = (ans * Pow(Dc[i], Dc[i + 1])) % mod;
        }
        return ans.ToString();
    }

    private static long Pow(long a, long b)
    {
        long mod = 998244353;
        long res = 1;
        for (long i = 0; i < b; i++)
        {
            res = (res * a) % mod;
        }
        return res;
    }


    private static string Read() { return Console.ReadLine(); }
    private static int ReadInt() { return int.Parse(Read()); }
    private static long ReadLong() { return long.Parse(Read()); }
    private static double ReadDouble() { return double.Parse(Read()); }
    private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
    private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
    private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
}