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.*;

public class Main {

	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		int n = sc.nextInt();
		long[] d = arrayLong(sc, n);
		sc.close();
		boolean can = true;
		if (d[0] != 0)
			can = false;
		Arrays.sort(d);
		long ans = 0;
		long mod = 998244353l;
		int prev = -1;
		int prevnum = 0;
		int prevnumprev = 0;
		// System.out.println(Arrays.toString(d));
		for (int i = 0; i < n; i++) {
			if (prev == -1) {
				if (d[i] == 0) {
					prev = 0;
					prevnum = 1;
				} else {
					break;
				}
			} else if (prev == 0) {
				if (d[i] == 0) {
					break;
				} else if (d[i] == 1) {
					prev = 1;
					ans = 1;
					prevnumprev = prevnum;
					ans *= prevnumprev;
				} else {
					break;
				}
			} else {
				if (d[i] == prev) {
					prevnum++;
					ans *= prevnumprev;
					ans %= mod;
				} else if (d[i] - 1 == prev) {
					prevnumprev = prevnum;
					prevnum = 1;
					ans *= prevnumprev;
					ans %= mod;
					prev++;
				} else {
					ans = 0;
					break;
				}
			}
		}
		System.out.println(can ? ans : 0);
	}

	public static int[] arrayInt(Scanner sc, int n) {
		int[] array = new int[n];
		for (int i = 0; i < n; i++) {
			array[i] = sc.nextInt();
		}
		return array;
	}

	public static long[] arrayLong(Scanner sc, int n) {
		long[] array = new long[n];
		for (int i = 0; i < n; i++) {
			array[i] = sc.nextLong();
		}
		return array;
	}

	public static double[] arrayDouble(Scanner sc, int n) {
		double[] array = new double[n];
		for (int i = 0; i < n; i++) {
			array[i] = sc.nextDouble();
		}
		return array;
	}
}