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: Wrong Answer)

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

public class Main {

	public static void main(String[] args) {
		// TODO 自動生成されたメソッド・スタブ
		Scanner sc = new Scanner(System.in);
		int n = sc.nextInt();
		int[] d = new int[n];
		int[] a = new int[n];
		for(int i = 0; i < n; i++){
			d[i] = sc.nextInt();
			a[i] = 0;
		}
		if(d[0] == 0){
			Arrays.sort(d);
			if(d[1] == 1){
				int j = 0;
				for(int k = 0; k < n; k++){
					if(d[k] == j){
						a[j]++;
					}
					else{
						if(d[k] == j+1){
							a[j+1]++;
							j++;
						}
						else{
							System.out.print(0);
							break;
						}
					}
				}
				long ans = 1;
				long tmp;
				for(int l = 1; l < n; l++){
					if(a[l] != 0){
						tmp = 1;
						for(int i = 0; i < a[l]; i++){
							tmp *= a[l-1];
							tmp %= 998244353;
						}
						ans *= tmp;
						ans %= 998244353;
					}
					else{
						System.out.print(ans);
						break;
					}
				}

			}
			else{
				System.out.print(0);
			}

		}
		else{
			System.out.print(0);
		}
	}

}