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 Rust (1.15.1) (judge verdict: Accepted)

use std::io::Read;
use std::collections::HashMap;

fn main() {
    let mut buf = String::new();
    std::io::stdin().read_to_string(&mut buf).unwrap();

    let answer = solve(&buf);

    println!("{}", answer);
}

fn solve(input: &str) -> String {
    let mut iterator = input.split_whitespace();

    let n: usize = iterator.next().unwrap().parse().unwrap();
    let d: Vec<usize> = (0..n)
        .map(|_| iterator.next().unwrap().parse().unwrap())
        .collect();
    if d[0] != 0 {
        return "0".to_string();
    }

    let mut counts: HashMap<usize, usize> = HashMap::new();
    for i in 0..n {
        *counts.entry(d[i]).or_insert(0) += 1;
    }

    if !counts.contains_key(&0) || *counts.get(&0).unwrap() != 1 {
        return "0".to_string();
    }

    let mut ans: usize = 1;
    let d_max: usize = *d.iter().max().unwrap();
    for i in 1..(d_max + 1) {
        if !counts.contains_key(&i) {
            return "0".to_string();
        }
        let a = counts.get(&(i - 1)).unwrap();
        let b = counts.get(&i).unwrap();
        ans = ans * mod_pow(*a, *b) % MOD
    }
    return ans.to_string();
}

const MOD: usize = 998244353;

fn mod_pow(a: usize, b: usize) -> usize {
    let mut result = 1;
    for _ in 0..b {
        result = result * a % MOD;
    }
    return result
}