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.Diagnostics;
using System.IO;
using System.Text;
using System.Linq;
using System.Collections.Generic;

class Program
{
    static void Main(string[] args) {
#if DEBUG
        //MyDebugger.MakeTestCase("../TextFile2.txt"); return;
        MyDebugger.Test(process, "../TextFile1.txt");
#else
        object ret = process(new StreamReader(Console.OpenStandardInput()));
        Console.WriteLine(ret);
#endif
    }
    
    static long mod = 998244353;

    static object process(TextReader input) {
        var n = int.Parse(input.ReadLine());
        var tbl = new long[n];
        long max = 0;

        var spl =input.ReadLine().Split(' ');
        for (int i = 0; i < n; i++) {
            var d = long.Parse(spl[i]);
            tbl[d]++;
            max = Math.Max(d, max);
        }

        if (spl[0] != "0") return 0;
        if (tbl[0] != 1) return 0;

        long sum = 1;

        for(int i=1; i<=max; i++) {
            if (tbl[i] == 0) return 0;

            long pre = tbl[i - 1];
            sum *= pow(pre, tbl[i]) % mod;
            //Debug.WriteLine(sum);
            sum %= mod;
        }
        return sum;
    }

    static long pow(long n, long m) {
        long ret = n;
        for(long i=1; i<m; i++) {
            ret *= n;
            ret %= mod;
        }
        return ret;
    }

    static long[] ToIntArray(string strs, int n) {
        var ret = new long[n];
        var spl =strs.Split(' ');
        for (long i = 0; i < n; i++)
            ret[i] = long.Parse(spl[i]);
        return ret;
    }
}