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.*;

public class Main{
	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		int N = sc.nextInt();
		int D[] = new int[N+1];
		for(int i=0;i<N;i++) D[i]=sc.nextInt();//initialized
		D[N]=N+100;
		if(D[0]!=0) {
			System.out.println(0);
			return;
		}
		Arrays.sort(D);
		long sum = 1;
		long cnt = 0;
		long bcnt = 0;
		long memo = -1;
		long waru = 998244353;
		for(int i=0;i<=N;i++) {
			if(memo==D[i])cnt++;
			else {
				if(memo+1!=D[i] && i!=0 && i!=N) {
					System.out.println(0);
					return;
				}
				long tmp = 1;
				for(int j=0;j<cnt+1;j++) {
					tmp *= (bcnt + 1);
					tmp%=waru;
				}
				//tmp = (long)Math.pow(bcnt+1,cnt+1);
				long memosum = sum;
				if(Long.MAX_VALUE/tmp<sum) {
				for(int j=1;j<tmp;j++) {
					sum += memosum;
					sum%=waru;
				}
				}else {
				sum *= tmp;
				sum%=waru;
				}
				bcnt = cnt;
				cnt=0;
			}
			memo = D[i];
		}
		
		System.out.println(sum%waru);
		
	}
}