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::str::FromStr;
use std::fmt::Debug;
use std::cmp;

#[allow(dead_code)]
fn read_as_vec<T>() -> Vec<T>
where T: FromStr, <T as FromStr>::Err : Debug{
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).unwrap();
    s.trim().split_whitespace().map(|c| T::from_str(c).unwrap()).collect()

}

fn pow_mod(x: u64, y: u64, mm: u64) -> u64{
    let mut ans = 1;
    for _ in 0..y{
        ans *= x;
        ans %= mm;
    }

    return ans;
}

fn main(){
    let n: usize = read_as_vec()[0];
    let d = read_as_vec::<usize>();
    let mm: u64 = 998244353;
    let mut count_table = vec![0;*d.iter().max().unwrap()+1];

    for i in 0..n{
        if i == 0 && d[i] != 0{
            println!("0");
            return;
        }

        if i != 0 && d[i] == 0{
            println!("0");
            return;
        }

        count_table[d[i]] += 1;
    }

    //if found zero
    if count_table.iter().any(|x| *x == 0){
        println!("0");
        return;
    }


    let mut ans = 1;
    for i in 1..count_table.len(){
        ans *= pow_mod(count_table[i-1], count_table[i], mm);
        ans %= mm;
    }

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