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.Linq;
using System.Collections.Generic;
using static System.Console;
using System.Text;
using System.IO;
using static System.Math;
namespace AtCoder
{
    class Program
    {
        static public long[] Sarray() { return ReadLine().Split().Select(long.Parse).ToArray(); }
        static public List<long> Slist() { return ReadLine().Split().Select(long.Parse).ToList(); }

        //static long Mod = (long)1e9 + 7;
        static void Main(string[] args)
        {
            //SetOut(new StreamWriter(OpenStandardOutput()) { AutoFlush = false });
            var Mod = 998244353L;
            var N = Sarray()[0];
            var Ai = Sarray();
            var dct = new Dictionary<long, long>();
            if (Ai[0] != 0)
            {
                WriteLine(0);
                return;
            }
            for (var i = 0; i < N; ++i)
            {
                if (!dct.ContainsKey(Ai[i])) dct[Ai[i]] = 0;
                ++dct[Ai[i]];
            }
            if ( 1 != dct[0])
            {
                WriteLine(0);
                return;
            }
            var max = Ai.Max();
            var ans = 1L;
            for (var i = 1; i <= max; ++i)
            {
                if (!dct.ContainsKey(i))
                {
                    WriteLine(0);
                    return;
                }
                var t = modpow(dct[i - 1], dct[i], Mod);
                ans = (ans * t) % Mod;
            }
            WriteLine(ans % Mod);
            //Out.Flush();
        }
        static long modpow(long a, long n, long mod)
        {
            var res = 1L;
            if (0 < n)
            {
                res = modpow(a, n / 2, mod);
                res = res * res % mod;
                if (0 != n % 2) res = res * a % mod;
            }
            return res;
        }
    }
}