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.Collections.Generic;
using System.Linq;

namespace AC
{
    class Program
    {
        static void Main(string[] args)
        {
            var N = ReadInt();
            var D = ReadArrayInt();
            var L = Enumerable.Repeat(new List<int>(), N).Select(l => new List<int>()).ToArray();
            for (int i = 0; i < N; i++) L[D[i]].Add(i + 1);
            if (L[0].Count() != 1)
            {
                Console.WriteLine(0);
                return;
            }
            else if (L[0][0] != 1)
            {
                Console.WriteLine(0);
                return;
            }
            var mod = 998244353L;
            var ans = 1L;
            var cnt = 1;
            var end = false;
            var zero = false;
            while (cnt < N)
            {
                if (end && L[cnt].Count != 0)
                {
                    zero = true;
                    break;
                }
                else if (end)
                {
                }
                else if (L[cnt].Count == 0)
                {
                    end = true;
                }
                else
                {
                    ans = ans * (long)Math.Pow(L[cnt - 1].Count(), L[cnt].Count());
                    ans %= mod;
                }
                cnt++;
            }

            Console.WriteLine(zero ? 0 : ans);
            Console.ReadKey();
        }

        static string Read() { return Console.ReadLine(); }
        static int ReadInt() { return int.Parse(Read()); }
        static long ReadLong() { return long.Parse(Read()); }
        static int[] ReadArrayInt() { return Read().Split(' ').Select(s => int.Parse(s)).ToArray(); }
        static long[] ReadArrayLong() { return Read().Split(' ').Select(s => long.Parse(s)).ToArray(); }
    }
}