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 namespace std;
typedef long long ll;
const ll MOD = 998244353;

int main() {
  int n;
  cin >> n;
  vector<int> cnts(n, 0);
  int ma = -1;
  bool is_valid = true;
  for (int i = 0; i < n; ++i) {
    int d;
    cin >> d;
    if (i == 0 && d != 0) is_valid = false;
    cnts[d]++;
    ma = max(d, ma);
  }

  if (!is_valid) {
    puts("0");
    return 0;
  }
  if (cnts[0] > 1) {
    puts("0");
    return 0;
  }
  ll ans = 1;
  for (int i = 0; i < ma; ++i) {
    if (cnts[i] == 0) {
      puts("0");
      return 0;
    }
    ll ad = (ll)cnts[i];
    for (int j = 0; j < cnts[i + 1] - 1; ++j) {
      ad *= (ll)cnts[i];
      ad %= MOD;
    }
    ans *= ad;
    ans %= MOD;
  }
  cout << ans << endl;
  return 0;
}