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.IO;
using System.Text;
using static System.Console;
using static System.Math;

namespace CP
{
    class Atria
    {
        static void Main(string[] args)
        {
            int n = int.Parse(ReadLine());
            int[] s = ReadLine().Split(' ').Select(int.Parse).ToArray();
            if (s[0] != 0)
            {
                WriteLine(0);
                return;
            }
            Array.Sort(s);
            var ls = new List<long>();
            long remain = s[0];
            long cnt = 1;
            for (int i = 1; i < n; i++)
            {
                if(Abs(remain - s[i]) >= 2 || s[i] == 0)
                {
                    WriteLine(0);
                    return;
                }
                if (remain == s[i])
                {
                    cnt++;
                }
                else
                {
                    ls.Add(cnt);
                    cnt = 1;
                    remain = s[i];
                }
            }
            ls.Add(cnt);
            long ans = 1;
            for (int i = 0; i < ls.Count-1; i++)
            {
                long tmp = 1;
                for (int j = 0; j < ls[i+1]; j++)
                {
                    tmp *= ls[i];
                    tmp %= 998244353;
                }
                ans *= tmp;
                ans %= 998244353;
            }
            WriteLine(ans);
        }

    }
}