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 <iostream>
#include <algorithm>
#include <vector>
#include <string>
#include <queue>
#include <set>
#include <map>
#include <functional>
#include <numeric>

using namespace std;
typedef long long int ll;
typedef pair <int,int> P;

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

int main() {
    ll n, ans = 1;
    vector<ll> d;
    ll e[100000];

    cin >> n;

    for (int i = 0; i < n; i++) {
         int temp;
         cin >> temp;
         d.push_back (temp);
    }

    if (d[0] != 0) {
        cout << 0 << endl;
        return 0;
    }

    sort(d.begin(), d.end());

    if (d[1] == 0) {
        cout << 0 << endl;
        return 0;
    }

    int j = 0;
    for (int i = 0; i < n; i++) {
        if (d[i] > j + 1) {
            cout << 0 << endl;
            return 0;
        } else if (d[i] == j + 1) {
            j++;
        }
        e[j]++;
    }

    for (int i = 0; i < n - 1; i++) {
        // cout << e[i] << endl;
        ans *= modpow(e[i], e[i+1], 998244353);
    }

    cout << ans << endl;
    
    return 0;


}