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 <iomanip>
#include <string>
#include <functional>
#include <utility>
#include <queue>
#include <vector>
#include <string>
#include <stack>
#define rep(i,n) for(ll i=0;i<n;i++)
using namespace std;
typedef long long int ll;
const ll  MOD = 998244353;
typedef pair<ll,ll> P;

ll N;
ll D[100100];
ll cnt[100100];

ll mod_pow(ll x, ll n){
  ll res = 1;
  while(n > 0){
    if(n % 2 == 1) res = (res * x )% MOD;
    x = (x * x) % MOD;
    n /= 2;
  }
  return res;
}

int main() {
  cin >> N;
  rep(i,N){
    cin >> D[i];
    cnt[D[i]]++;
  }
  if (D[0] != 0) {
    cout << "0" << endl;
    return 0;
  }
  
  ll sum = 0;
  ll ans = 1;
  rep(i,N){
    if(sum < N && cnt[i] == 0) {
      cout << "0" << endl;
      return 0;
    }
    if(i == 0 && cnt[i] != 1) {
      cout << "0" << endl;
      return 0;
    }
    
    sum += cnt[i];
    if (i != 0) {
      ans = (ans * mod_pow(cnt[i-1], cnt[i])) %MOD;
    }
    
    
    if (sum == N) break;
  }
  cout << ans << endl;
}