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<iostream>
#include<vector>
#include<string>
#include<algorithm>
#include<tuple>
#include<map>
#include<queue>
#include<math.h>
#include<iomanip>
#define rep(i, n) rep2(i,0,n)
#define rep2(i, s, n) for (int i = (s); i < (int)(n); i++)
#define PI acos(-1);
typedef long long ll;
using namespace std;
ll Mod = 998244353;
//scanf("%d", &a)

int main() {
	int n;
	cin >> n;
	vector<int> d(n);
	rep(i, n) scanf("%d", &d[i]);
	if (d[0] != 0) { 
		cout << 0 << endl; return 0;
	}
	sort(d.begin(), d.end());
	rep2(i, 1, n) {
		if (d[i] == 0|| d[i] - d[i - 1] > 1) { cout << 0 << endl; return 0; }
	}

	ll ans = 1;
	vector<int> s(n, 0);
	int j = 0;
	int count = 1;
	
	rep2(i, 2, n) {
		if (d[i] != d[i - 1]) {
			s[j] = count;
			++j;
			count = 1;
		}
		else count++;
		if (i == n - 1) s[j] = count;
	}
	rep2(i, 1, n) {
		if (s[i] == 0) break;
		//ans = ans *ll(powl(s[i - 1], s[i]))% Mod;
		rep(j, s[i]) {
			ans *= s[i - 1];
			ans %= Mod;
		}
	}
	cout << ans << endl;
}