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 static System.Console;
using static System.Math;

namespace B_NK2
{
    class MainClass
    {

        static int[] n;
        static long[] d;

        public static void Main(string[] args)
        {
            long div = 998244353;
            n = Array.ConvertAll(ReadLine().Split().ToArray(), int.Parse);
            d = Array.ConvertAll(ReadLine().Split().ToArray(), long.Parse);
            var dl = d.ToList();

            if (dl[0] != 0)
            {
                WriteLine(0);
                return;
            }
            dl.RemoveAt(0);
            dl.Sort();
            if (dl[0] != 1)
            {
                WriteLine(0);
                return;
            }
            long ans = 1;
            long dist = 1;

            long beforecnt = 1;
            long nextcnt = 0;

            for (int i = 0; i < n[0] - 1; i++)
            {
                if (dl[i] != dist && dl[i] != dist + 1)
                {
                    WriteLine(0);
                    return;
                }
                else if (dl[i] == dist)
                {
                    ans = ans * beforecnt % div;
                    nextcnt++;
                }
                else
                {
                    beforecnt = nextcnt;
                    nextcnt = 1;
                    dist = dl[i];
                    ans = ans * beforecnt % div;
                }


            }
            WriteLine(ans);
        }
    }
}