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> // cout, endl, cin
#include <string> // string, to_string, stoi
#include <vector> // vector
#include <algorithm> // min, max, swap, sort, reverse, lower_bound, upper_bound
#include <utility> // pair, make_pair
#include <tuple> // tuple, make_tuple
#include <cstdint> // int64_t, int*_t
#include <cstdio> // printf
#include <map> // map
#include <queue> // queue, priority_queue
#include <set> // set
#include <stack> // stack
#include <deque> // deque
#include <unordered_map> // unordered_map
#include <unordered_set> // unordered_set
#include <bitset> // bitset
#include <cctype> // isupper, islower, isdigit, toupper, tolower
#include<cmath>
#include<limits>
#define ll long long
#define ALL(x) (x).begin(),(x).end()
#define rep(i, n) for(int i = 0; i < (int)(n); i++)
#define int_INF 2147483647
#define pint_INF 2000000000
#define ll_INF 	9223372036854775807
#define MOD 1000000007
#define vi vector<int>
#define vvi vector<vector<int>>
#define vvc vector<vector<char>>
#define vll vector<long long>
#define pii pair<int,int>
#define vpii vector<pair<int,int>>
#define sysp system("PAUSE")
using namespace std;
#define int ll

ll pow(ll x, ll n) {
    ll res = 1;
    rep(i, n) res *= x;

    return res;
}
signed main() {
    int N; cin >> N;
    vi table(N);

    rep(i, N) {
        int x; cin >> x; table[x]++;
    }

    ll ans = 1;

    if (table[0] == 0) {
        ans = 0;
    }
    else {
        int index = 0;
        int sum = 0;
        for (; index < N; index++) {
            sum += table[index];
            if (table[index] == 0) {
                break;
            }
        }

        for (int i = 1; i < N; i++) {
            ans *= pow(table[i - 1], table[i]);
        }

        if (sum != N) {
            ans = 0;
        }
    }

    cout << ans << endl;
    sysp;
}