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.HashMap;
import java.util.Map;
import java.util.Scanner;
import java.util.TreeSet;

public class Main {

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

        Map<Integer, Integer> map = new HashMap<>();

        int first = sc.nextInt();
        map.put(first, 1);
        if (first != 0) {
            impossible = true;
        }

        for (int i = 1; i < n; i++) {
            int tmp = sc.nextInt();
            if (map.containsKey(tmp)) {
                map.put(tmp, map.get(tmp) + 1);
            } else {
                map.put(tmp, 1);
            }
        }
        // for debug
        // System.out.println(map);

        // 0が 0 or 複数個はNG
        if (!impossible && map.get(0) != 1) {
            impossible = true;
        }

        // keyが0からの連続数でない場合はNG
        TreeSet<Integer> tree = new TreeSet<>(map.keySet());
        if (tree.size() - 1 != tree.last()) {
            impossible = true;
        }
        // for debug
        // System.out.println(tree);

        int ans = 1;
        if (impossible) {
            ans = 0;
        } else {
            for (int i = 2; i <= tree.last(); i++) {
                // ans *= ((map.get(i - 1) ^ map.get(i)) % 998244353);
                for (int j = 0; j < map.get(i); j++) {
                    ans *= map.get(i - 1) % 998244353;
                }
            }
        }
        System.out.println(ans);

        sc.close();
    }

}