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;
long long mod = 998244353;
long long modpow(long long a, long long n) {
    long long res = 1;
    while (n > 0) {
        if (n & 1) res = res * a % mod;
        a = a * a % mod;
        n >>= 1;
    }
    return res;
}
int main()
{
    long long n;
    cin >> n;
    vector< long long > d(n);
    vector< long long > num(n, 0);
    for (long long i = 0; i < n; i++) {
        cin >> d[i];
        if (d[i] == 0 && i != 0) {
            cout << 0 << endl;
            return 0;
        }
        num[d[i]]++;
    }
    long long ans = 1;
    for (int i = 0; i < n - 1; i++) {
        ans = (ans * modpow(num[i], num[i + 1])) % mod;
    }
    // for (int i = 0; i < n; i++) {
    //     cout << num[i] << " ";
    // }
    // cout << endl;
    cout << ans << endl;
    return 0;
}