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(non_snake_case)]
#![allow(dead_code)]
#![allow(unused_macros)]
#![allow(unused_imports)]

use std::str::FromStr;
use std::io::*;
use std::collections::*;
use std::cmp::*;

struct Scanner<I: Iterator<Item = char>> {
    iter: std::iter::Peekable<I>,
}

macro_rules! exit {
    () => {{
        exit!(0)
    }};
    ($code:expr) => {{
        if cfg!(local) {
            writeln!(std::io::stderr(), "===== Terminated =====")
                .expect("failed printing to stderr");
        }
        std::process::exit($code);
    }}
}

impl<I: Iterator<Item = char>> Scanner<I> {
    pub fn new(iter: I) -> Scanner<I> {
        Scanner {
            iter: iter.peekable(),
        }
    }

    pub fn safe_get_token(&mut self) -> Option<String> {
        let token = self.iter
            .by_ref()
            .skip_while(|c| c.is_whitespace())
            .take_while(|c| !c.is_whitespace())
            .collect::<String>();
        if token.is_empty() {
            None
        } else {
            Some(token)
        }
    }

    pub fn token(&mut self) -> String {
        self.safe_get_token().unwrap_or_else(|| exit!())
    }

    pub fn get<T: FromStr>(&mut self) -> T {
        self.token().parse::<T>().unwrap_or_else(|_| exit!())
    }

    pub fn vec<T: FromStr>(&mut self, len: usize) -> Vec<T> {
        (0..len).map(|_| self.get()).collect()
    }

    pub fn mat<T: FromStr>(&mut self, row: usize, col: usize) -> Vec<Vec<T>> {
        (0..row).map(|_| self.vec(col)).collect()
    }

    pub fn char(&mut self) -> char {
        self.iter.next().unwrap_or_else(|| exit!())
    }

    pub fn chars(&mut self) -> Vec<char> {
        self.get::<String>().chars().collect()
    }

    pub fn mat_chars(&mut self, row: usize) -> Vec<Vec<char>> {
        (0..row).map(|_| self.chars()).collect()
    }

    pub fn line(&mut self) -> String {
        if self.peek().is_some() {
            self.iter
                .by_ref()
                .take_while(|&c| !(c == '\n' || c == '\r'))
                .collect::<String>()
        } else {
            exit!();
        }
    }

    pub fn peek(&mut self) -> Option<&char> {
        self.iter.peek()
    }
}

use std::ops::*;
use std::fmt;

#[derive(Copy, Clone, PartialEq, Debug)]
struct ModInt(usize);

const MOD: usize = 998244353;

impl ModInt {
    fn pow(self, power: usize) -> Self {
        if power == 0 {
            return ModInt(1);
        }
        if power % 2 == 0 {
            let t = self.pow(power/2);
            return t * t;
        }
        self * self.pow(power-1)
    }

    fn inv(self) -> Self {
        self.pow(MOD - 2)
    }
}

impl Add for ModInt {
    type Output = Self;

    fn add(self, rhs: Self) -> Self {
        let mut tmp = self;
        tmp.0 += rhs.0;
        if tmp.0 >= MOD {
            tmp.0 -= MOD;
        }
        tmp
    }
}

impl AddAssign for ModInt {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl Sub for ModInt {
    type Output = Self;

    fn sub(self, rhs: Self) -> Self {
        let mut ret = self;
        if ret.0 < rhs.0 {
            ret.0 += MOD;
        }
        ret.0 -= rhs.0;
        ret
    }
}

impl SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

impl Mul for ModInt {
    type Output = Self;

    fn mul(self, rhs: ModInt) -> Self {
        let mut ret = self;
        ret.0 *= rhs.0;
        ret.0 %= MOD;
        ret
    }
}

impl MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl Div for ModInt {
    type Output = ModInt;

    fn div(self, rhs: ModInt) -> Self {
        let mut ret = self;
        ret.0 *= rhs.inv().0;
        ret.0 %= MOD;
        ret
    }
}

impl DivAssign for ModInt {
    fn div_assign(&mut self, rhs: Self) {
        *self = *self / rhs;
    }
}

impl fmt::Display for ModInt {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        write!(f, "{}", self.0)
    }
}

fn main() {
    let cin = stdin();
    let cin = cin.lock();
    let mut sc = Scanner::new(cin.bytes().map(|c| c.unwrap() as char));
    let N: usize = sc.get();
    let mut D: Vec<usize> = sc.vec(N);
    if D[0] != 0 {
        println!("0");
        return;
    }
    let D = &mut D[1..];
    D.sort();
    let mut cnt = vec![0; N];
    cnt[0] = 1;
    let mut ans = ModInt(1);
    for i in 0..D.len() {
        if D[i] == 0 {
            println!("0");
            return;
        }
        ans *= ModInt(cnt[D[i] - 1]);
        cnt[D[i]] += 1;
    }
    println!("{}", ans);
}