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.Linq;

class Program
{
    static void Main(string[] args)
    {
        int N = int.Parse(Console.ReadLine());
        string[] input = Console.ReadLine().Split();
        int[] D = new int[N];
        var r = 1L;
        for (int i = 0; i < N; i++)
        {
            D[i] = int.Parse(input[i]);
            if (D[0] > 0) { r = 0; }
            if (i!=0&&D[i] == 0) { r = 0; }
        }
        Array.Sort(D);
        var c = D.GroupBy(x => x).ToDictionary(g => g.Key, g => g.Count());
        var max = c.Keys.Max();
        if (c.Count <= max) { Console.WriteLine(0); return; }
        Console.WriteLine(c[0]);
        Console.WriteLine(max);
        for (int i = 0; i < max; i++) r = MMul(r, MPow(c[i], c[i + 1]));
        Console.WriteLine(r);
    }
    const int M = 998244353;
    static long MPow(long b, int i)
    {
        for (var r = 1L; ; b = b * b % M)
        {
            if (i % 2 > 0) r = r * b % M;
            if ((i /= 2) < 1) return r;
        }
    }
    static long MMul(long x, long y) => x * y % M;
}