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>
#include <iostream>
#include <vector>
#include <string>
#include <cmath>
#include <algorithm>
#include <map>
#include <iomanip>
#include <stdlib.h>
#include <queue>

using namespace std;

const int nmax = 1e9 + 7;
const long long INF = 1e18;
const double PI = 2 * asin(1);
typedef long long ll;

ll AN(ll A, ll N, ll mod){
  if (N == 0){
    return 1;
  }else{
    if (N % 2 == 0){
      return AN(A * A % mod, N / 2, mod);
    }else{
      return A * AN(A * A % mod, N / 2, mod) % mod;
    }
  }
}

int main(){
  int N; cin >> N;
  
  ll ans = 1;
  vector <ll> D(N);
  for (int i = 0; i < N; i++){
    cin >> D[i];
  }
  if (D[0] != 0) ans = 0;

  sort(D.begin(), D.end());
  vector <ll> cnt(D[N - 1] + 1);
  for (int i = 0; i < N; i++){
    cnt[D[i]]++;
  }

  if (cnt[0] != 1) ans = 0;
  
  for (int i = 1; i < D[N - 1] + 1; i++){
    ans *= AN(cnt[i - 1], cnt[i], 998244353);
    ans %= 998244353;
  }
  cout << ans << endl;
  
  return 0;
}