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: Wrong Answer)

#include <bits/stdc++.h>
#define rep(i, n) for (int i = 0; i < (n); ++i)
using namespace std;
 
int main() {
  int N;
  long ans;
  ans = 1;
  bool a = false;
  cin >> N;
  if (N == 1) {
    a = true;
    ans = 0;
  }
  int D[N + 1];
  int count[N + 1];
  for (int i = 0; i < N; ++i) {
    cin >> D[i];
    count[i] = 0;
    if (D[0] != 0) {
      a = true;
      break;
    }
  }
  if (N == 1 && D[0] == 0) {
    a = true;
    ans = 1;
  }
  for (int i = 1; i < N; ++i) {
    ++count[D[i]];
    if (D[i] == 0) {
      a = true;
      break;
    }
  }
  for (int i = 1; i < N - 1; ++i) {
    if (a) {
      ans = 0;
      break;
    }
    int num1 = count[i];
    int num2 = count[i+1];
    if (num2 == 0) {
      continue;
    }
    ans *= pow(num1, num2);
  }
  ans = ans % 998244353;
  cout << ans <<endl;
  return 0;
}