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.Numerics;
using static System.Console;
using static System.Math;

class Program
{
    const long mod = 998244353;
    static void Main()
    {
        var N = int.Parse(ReadLine());
        var D = ReadLine().Split().Select(int.Parse).ToArray();
        if (D[0] != 0)
        {
            WriteLine(0);
            return;
        }
        var zero = false;
        for (int i = 1; i < N; i++)
        {
            if (D[i] == 0) zero = true;
        }
        if (zero)
        {
            WriteLine(0);
            return;
        }
        Array.Sort(D);
        var ren = true;
        for (int i = 1; i < N; i++)
        {
            if (D[i] - D[i - 1] > 1) ren = false;
        }
        if (!ren)
        {
            WriteLine(0);
            return;
        }

        var d = new int[N];
        for (int i = 0; i < N; i++)
        {
            d[D[i]]++;
        }

        var m = D.Max();
        var ans = 1L;
        for (int i = 0; i < m; i++)
        {
            ans *= ModPow(d[i] , d[i + 1] , mod);
            ans %= mod;
        }
        WriteLine(ans);
    }

    public static long ModPow(long x, long n, long m)
    {
        var res = 1L;
        while (n != 0)
        {
            if ((n & 1) == 1) res = res * x % m;
            x = x * x % m;
            n >>= 1;
        }
        return res;
    }
}