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.Scanner;

public class Main {

    static Scanner sc = new Scanner(System.in);
    static final long MOD = 998244353;

    public static void main(String[] args) {
        int N = sc.nextInt();
        if (sc.nextInt() != 0) {
            System.out.println(0);
            return;
        }
        int max = 0;
        int[] count = new int[N];
        for (int i = 1; i < N; i++) {
            int d = sc.nextInt();
            count[d]++;
            max = Math.max(max, d);
        }
        if (count[0] != 0) {
            System.out.println(0);
            return;
        }
        long ans = 1;
        for (int i = 2; i <= max; i++) {
            for (int j = 0; j < count[i]; j++) {
                ans *= count[i - 1];
                ans %= MOD;
            }
        }
        System.out.println(ans);
    }

}