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)

#![allow(unused)]
use std::io::Write;
use std::ops::{Add, Sub, Mul};

#[allow(unused)]
macro_rules! debug {
    ($($format:tt)*) => (write!(std::io::stderr(), $($format)*).unwrap());
}
#[allow(unused)]
macro_rules! debugln {
    ($($format:tt)*) => (writeln!(std::io::stderr(), $($format)*).unwrap());
}

const M: isize = 998244353;

fn main() {
    let n: usize = {
        let mut s = String::new();
        std::io::stdin().read_line(&mut s).unwrap();
        let s = s.trim_right().to_owned();
        s.parse().unwrap()
    };

    let d: Vec<usize> = {
        let mut s = String::new();
        std::io::stdin().read_line(&mut s).unwrap();
        let s = s.trim_right().to_owned();
        s.split_whitespace().map(|x| x.parse().unwrap()).collect()
    };

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

    let mut e = vec![0; n];
    let mut max_e = 0;
    for a in d {
        e[a] += 1;
        if a > max_e {
            max_e = a;
        }
    }

    // debugln!("{:?}", e);
    // debugln!("{}", max_e);

    if e[0] != 1 {
        println!("{}", 0);
        return;
    }

    let mut res = Modulo(1);
    for i in 1..(max_e + 1) {
        res = res * Modulo(e[i - 1]).pow(e[i] as usize);
    }

    println!("{}", res.0);
}

#[derive(Debug, Copy, Clone)]
struct Modulo(isize);

impl Modulo {
    fn pow(self, p: usize) -> Modulo {
        if self == Modulo(0) {
            return Modulo(0);
        }
        if p == 0 {
            return Modulo(1);
        }

        if p % 2 == 0 {
            let half = self.pow(p / 2);
            half * half
        } else {
            self.pow(p - 1) * self
        }
    }
}

impl Add for Modulo {
    type Output = Modulo;

    fn add(self, other: Modulo) -> Modulo {
        Modulo((self.0 + other.0) % M)
    }
}

impl Sub for Modulo {
    type Output = Modulo;

    fn sub(self, other: Modulo) -> Modulo {
        let mut value = (self.0 - other.0) % M;
        if value < 0 {
            value += M;
        }
        Modulo(value)
    }
}

impl Mul for Modulo {
    type Output = Modulo;

    fn mul(self, other: Modulo) -> Modulo {
        Modulo((self.0 * other.0) % M)
    }
}

impl PartialEq for Modulo {
    fn eq(&self, &other: &Self) -> bool {
        self.0 == other.0
    }
}

impl Eq for Modulo {}