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++14 (GCC 5.4.1) (judge verdict: Accepted)


#include <iostream>
#include <algorithm>
#include <cmath>
#include <queue>
#include <set>
#include <map>
#include <vector>
#include <string>
using namespace std;

typedef long long ll;
typedef long double ld;

const int M=1e5+5;
const ll mod=998244353;
ll ct[M];
int main()
{
	ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);
	int mx=0;
	int n;
	cin>>n;
	for(int i=1;i<=n;i++)
	{
		int d;
		cin>>d;
		if(i==1 && d)
		{
			cout<<"0\n";
			return 0;
		}
		if(i>1 && d==0)
		{
			cout<<"0\n";
			return 0;
		}
		ct[d]++;
		mx=max(mx,d);
	}
	ll res=1;
	for(ll i=1;i<=mx;i++)
	{
		if(ct[i]==0)
			res=0;
		for(int j=1;j<=ct[i];j++)
			res=(res*ct[i-1])%mod;
	}
	cout<<res<<"\n";
	return 0;
}