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>
#define int long long
#define mod (int)(1e9+7)
#define inf (int)(3e18)
#define rep(i,n) for(int i=0;i<n;i++)
#define REP(i,n) for(int i=1;i<n;i++)
#define P std::pair<int,int>
#define PiP std::pair<int,std::pair<int,int>>
#define all(v) v.begin(),v.end()
#define mkp std::make_pair
#define prique(T) std::priority_queue<T,vector<T>,greater<T>>
using namespace std;
template<class T> inline void chmax(T& a, T b) { a = std::max(a, b); }
template<class T> inline void chmin(T& a, T b) { a = std::min(a, b); }

bool prime(int x) {
	for (int i = 2; i * i <= x; i++) {
		if (x % i == 0)return false;
	}
	return x != 1;
}
int gcd(int x, int y) {
	if (y == 0)return x;
	return gcd(y, x % y);
}
int lcm(int x, int y) {
	return x / gcd(x, y) * y;
}
int kai(int x, int y) {
	int res = 1;
	for (int i = x - y + 1; i <= x; i++) {
		res *= i; res %= mod;
	}
	return res;
}
int mod_pow(int x, int y, int m) {
	int res = 1;
	while (y > 0) {
		if (y & 1) {
			res = res * x % m;
		}
		x = x * x % m;
		y >>= 1;
	}
	return res;
}
int comb(int x, int y) {
	if (y > x)return 0;
	return kai(x, y) * mod_pow(kai(y, y), mod - 2, mod) % mod;
}
int get_rand(int MIN, int MAX) {
	std::random_device rnd;
	std::mt19937 mt32(rnd());
	std::uniform_int_distribution<int>engine(MIN, MAX);
	return engine(mt32);
}
/*--------Library Zone!--------*/

int n;
int cnt[114514];
signed main() {
	cin >> n;
	rep(i, n) {
		int a; cin >> a; cnt[a]++;
		if (i == 0 && a != 0) {
			cout << 0 << endl; return 0;
		}
	}
	if (cnt[0] != 1) {
		cout << 0 << endl; return 0;
	}
	bool flag = false;
	rep(i, n) {
		if (cnt[i] == 0)flag = true;
		if (flag && cnt[i]) {
			cout << 0 << endl; return 0;
		}
	}
	int ans = 1;
	REP(i, n) {
		ans *= mod_pow(cnt[i - 1], cnt[i], 998244353);
		ans %= 998244353;
	}
	cout << ans << endl;
}