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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.util.Arrays;
import java.util.Scanner;

public class Main {
	private static Scanner sc;

	public static void main(String[] args) {
		sc = new Scanner(System.in);
		int n = sc.nextInt();

		int[] array = new int[n];
		int[] counts = new int[n];

		for (int i = 0; i < n; i ++) {
			array[i] = sc.nextInt();
		}

		if (array[0] != 0) {
			System.out.println(0);
			System.exit(0);
		}

		Arrays.sort(array);

		long result = 1;
		long beforeCount = 1;
		long count = 0;
		long num = 0;

		if (array[0] != 0 || array[1] == 0) {
			result = 0;
		} else {
			for (int i : array) {
				if (i != num) {
					if (i == num + 1) {
						num += 1;
						beforeCount = count;
						count = 0;
					} else {
						result = 0;
						break;
					}
				}
				count ++;

				long ibc = beforeCount % 998244353;

				result = (result * ibc % 998244353);


			}
		}

		System.out.println(result);
	}

}