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;

class B
{
    static void Main()
    {
#if DEBUG
        string exampleInput =
        @"4
1 1 1 1
";
        Console.SetIn(new System.IO.StringReader(exampleInput));
#endif

        long MOD = 998244353;

        var N = int.Parse(Console.ReadLine());
        var D = Console.ReadLine().Split(' ').Select(int.Parse).ToList();
        if (D[0] != 0) { Console.WriteLine(0); Console.Out.Flush(); return; }
        int next = 1;
        int prev = 1;
        long result = 1;
        foreach(var g in D.Skip(1).GroupBy(d => d).OrderBy(g => g.Key))
        {
            if(g.Key != next) { Console.WriteLine(0); Console.Out.Flush(); return; }
            foreach(var gg in g) result = (result * prev) % MOD;
            prev = g.Count();
            next++;
        }
        Console.WriteLine(result);
        Console.Out.Flush();
    }
}