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;

namespace B
{
    class Program
    {
        static void Main(string[] args)
        {
            Console.WriteLine(solver());
        }
        static long solver()
        {
            var N = int.Parse(Console.ReadLine());
            var D = Array.ConvertAll(Console.ReadLine().Split(' '), int.Parse);
            long MOD = 998244353;

            if (D[0] != 0) return 0;

            var distance = new long[D.Max() + 1];
            foreach (int d in D) distance[d] += 1;
            if (distance[0] != 1) return 0;
            long result = 1;
            for (int i = 1; i < distance.Length; i++)
            {
                for (int j = 0; j < distance[i]; j++)
                {
                    result *= distance[i - 1];
                    result %= MOD;
                }
            }
            return result;
        }
    }
}