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: Wrong Answer)

using System;
using System.IO;
using System.Linq;
using System.Collections;
using System.Collections.Generic;
using System.Text;
using System.Threading.Tasks;
using System.Text.RegularExpressions;
using Debug = System.Diagnostics.Debug;
using System.Runtime.CompilerServices;
using MethodImplOptions = System.Runtime.CompilerServices.MethodImplOptions;
using MethodImplAttribute = System.Runtime.CompilerServices.MethodImplAttribute;

class Program
{
    static void Main(string[] args)
    {
        long N = long.Parse(Console.ReadLine());
        string []input = Console.ReadLine().Split();
        long[] D = new long[N];
        for(long i = 0; i < N; i++)
        {
            D[i] = long.Parse(input[i]);
        }
        Array.Sort(D);
        ModInt ans = 1;
        ModInt count = 1;
        ModInt temp =1;
        for (long i = 1; i < N; i++)
        {
            if (D[i]==0 || D[0] != 0) { ans = 0; }
            if (i>1&&D[i - 1] != D[i]) {
                ans =ans * (Power(temp, (i - count)) % 998244353) % 998244353;
                temp = i - count;
                count =i;
            }
            
        }
        ans = ans * (Power(temp, (N - count)) % 998244353) % 998244353;
        Console.WriteLine(ans% 998244353);
    }
    static ModInt Power(ModInt n, long m)
    {
        ModInt pow = n;
        ModInt res = 1;
        while (m > 0)
        {
            if ((m & 1) == 1) res *= pow;
            pow *= pow;
            m >>= 1;
        }
        return res;
    }
    static List<ModInt> factorialMemo = new List<ModInt>() { 1 };
    static ModInt Factorial(int x)
    {
        for (int i = factorialMemo.Count; i <= x; i++) factorialMemo.Add(factorialMemo.Last() * i);
        return factorialMemo[x];
    }
    static List<ModInt> factorialInverseMemo = new List<ModInt>() { 1 };
    static ModInt FactorialInverse(int x)
    {
        for (int i = factorialInverseMemo.Count; i <= x; i++) factorialInverseMemo.Add(factorialInverseMemo.Last() / i);
        return factorialInverseMemo[x];
    }
}

struct ModInt
{
    const int MOD = 998244353;
    long Data;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public ModInt(long data) { if ((Data = data % MOD) < 0) Data += MOD; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static implicit operator long(ModInt modInt) => modInt.Data;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static implicit operator ModInt(long val) => new ModInt(val);
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator +(ModInt a, ModInt b)
    {
        long res = a.Data + b.Data;
        return new ModInt() { Data = res >= MOD ? res - MOD : res };
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator +(ModInt a, long b) => a.Data + b;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator -(ModInt a, ModInt b)
    {
        long res = a.Data - b.Data;
        return new ModInt() { Data = res < 0 ? res + MOD : res };
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator -(ModInt a, long b) => a.Data - b;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator *(ModInt a, ModInt b) => new ModInt() { Data = a.Data * b.Data % MOD };
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator /(ModInt a, ModInt b) => a.Data * GetInverse(b);

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public override string ToString() => Data.ToString();

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    static long GetInverse(long a)
    {
        long div;
        long p = MOD;
        long x1 = 1, y1 = 0, x2 = 0, y2 = 1;
        while (true)
        {
            if (p == 1) return x2;
            div = a / p;
            x1 -= x2 * div;
            y1 -= y2 * div;
            a %= p;
            if (a == 1) return x1;
            div = p / a;
            x2 -= x1 * div;
            y2 -= y1 * div;
            p %= a;
        }
    }
}