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 Java (OpenJDK 11.0.6) (judge verdict: Wrong Answer)

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;

public class Main {

	public static void main(String[] args) throws IOException {
		BufferedReader br = new BufferedReader(new InputStreamReader(System.in));

		int n = Integer.parseInt(br.readLine());

		String str[] = br.readLine().split(" ");
		if (!str[0].equals("0")) {
			System.out.println("0");
			return;
		}

		int s[] = new int[100010];
		int max = 0;

		for (int i = 0; i < n; i++) {
			int d = Integer.parseInt(str[i]);
			s[d]++;
			max = Math.max(max, d);
		}

		if (s[0] != 1) {
			System.out.println("0");
			return;
		}

		long ans = 1;
		for (int i = 1; i < max; i++) {
			if (s[i] == 0) {
				System.out.println("0");
				return;
			}
			ans *= modpow(s[i], s[i + 1], 998244353) % 998244353;
		}

		System.out.println(ans % 998244353);
	}

	// a^n mod を計算する
	static long modpow(long a, long n, long mod) {
		long res = 1;
		while (n > 0) {
			if ((n & 1) != 0)
				res = res * a % mod;
			a = a * a % mod;
			n >>= 1;
		}
		return res;
	}
}