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 <bits/stdc++.h>
typedef long long ll;
#define  rep(i,a,b)for(long long i=a;i<b;++i)
#define  rrep(i,a,b)for(int i=a;i>=b;--i)
#define  yesno(flag)if(flag)cout<<"Yes"<<endl; else cout<<"No"<<endl;
using namespace std;
const ll INF=1000000000000LL;
const ll r197=1000000007;
typedef pair<ll,ll>P;
typedef pair<int,P>Pa;
bool Inframe(int a,int b,int c,int d,int h,int w){
	if(a+b>=0&&a+b<h&&c+d>=0&&c+d<w)return true;
	return false;
}
vector<int> v[601];
int n;
ll ans=1;
string s;
multiset<ll> lset,rset;
ll d[100005];
ll a[100005]={};
ll myPow(ll x, ll m){
	x%=998244353;
  if(m== 0)
    return 1;
  if(m % 2 == 0)
    return (myPow((x * x)%998244353, m / 2))%998244353;
  else
    return (x * myPow(x, m - 1))%998244353;
}
int main() {
	cin>>n;
	ll ma=0;
	rep(i,0,n){
		cin>>d[i];
		a[d[i]]++;
		ma=max(ma,d[i]);
	}
	if(a[0]!=1||d[0]!=0){
		cout<<0<<endl;
		return 0;
	}
	rep(i,1,ma+1){
		ans=(ans*myPow(a[i-1],a[i]))%998244353;

	}
	cout<<ans<<endl;
	return 0;
}