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 Java (OpenJDK 11.0.6) (judge verdict: Accepted)

import java.util.*;

public class Main {

    static int mod = 998244353;
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();
        int[] d  = new int[n];
        boolean zeroStart = false;
        int max = 0;
        for (int i = 0; i < n; i++) {
            int x = sc.nextInt();
            d[x]++;
            if(i == 0 && x == 0) zeroStart = true;
            if(max < x) max = x;
        }
        long ans = zeroStart && d[0] == 1 ? 1L : 0L;
        for (int i = 1; i <= max; i++) {
            ans = ans * modPow((long)d[i-1], d[i]) %mod;
        }
        System.out.println(ans);
        sc.close();

    }
    public static long modPow(long base, int exp) {
        long ret = 1;
        while(exp > 0){
            if((exp & 1) == 1){
                ret = ret * base %mod;
            }
            base = base * base %mod;
            exp >>= 1;
        }
        return ret;
    }

}