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>
using ll = long long;
constexpr int MOD = 998244353;
constexpr int DEBUG = 0;
using namespace std;

ll powMod(ll a, ll n, ll mod) {
    ll res = 1;
    while (n > 0) {
        if (n & 1) res = res * a % mod;
        a = a * a % mod;
        n >>= 1;
    }
    return res;
}

int main() {
  int n;
  cin >> n;
  int maxD = 0;
  vector<int> num(n, 0);
  for (int i = 0; i < n; i++) {
    int di;
    cin >> di;
    num[di]++;
    if (maxD <= di)
      maxD = di;
    
    if (i != 0 && di == 0) {
      cout << 0 << endl;
      return 0;
    }
  }
  if (num[0] != 1) {
    cout << 0 << endl;
    return 0;
  }
  ll total = 1;
  for (int i = 0; i < maxD; i++) {
    total *= powMod(num[i], num[i+1], MOD);
    total %= MOD;
    if (DEBUG == 1) {
      ll k = powMod(num[i], num[i+1], MOD) % MOD;
      cout << total << ":" << num[i] << "^" << num[i+1] << "=" <<  k << endl;
    }
  }
  cout << total << endl;

  return 0;
}