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::{stdin, Read, StdinLock};
use std::str::FromStr;

fn main() {
    let cin = stdin();
    let mut scan = Scanner::new(cin.lock());
    let n = scan.read::<usize>();
    let d = scan.readn::<usize>(n);
    let mut cnt = vec![0;n];
    for i in 0..n {
        cnt[d[i]] += 1;
    }
    let mut ret = if cnt[0] == 1 && d[0] == 0 {Co(1)} else {Co(0)};
    for i in 2..n {
        ret *= Co(cnt[i-1]).pow(cnt[i]);
    }
    println!("{}", ret);
}

#[allow(dead_code)]
const M: i64 = 998244353;

// region modulo
#[allow(dead_code)]
use std::ops::*;

#[allow(dead_code)]
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub struct Co (pub i64);

#[allow(dead_code)]
impl Co {
    pub fn pow(self, mut n: i64) -> Self {
        let mut a = self;
        let mut res = Co(1);
        while n > 0 {
            if n & 1 == 1 {
                res *= a;
            }
            a = a * a;
            n >>= 1;
        }
        res
    }
}

#[allow(dead_code)]
impl std::fmt::Display for Co {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

#[allow(dead_code)]
impl From<i32> for Co {
    fn from(value: i32) -> Self {
        Co((value as i64 % M + M) % M)
    }
}
#[allow(dead_code)]
impl From<i64> for Co {
    fn from(value: i64) -> Self {
        Co((value as i64 % M + M) % M)
    }
}
#[allow(dead_code)]
impl From<usize> for Co {
    fn from(value: usize) -> Self {
        Co((value as i64 % M + M) % M)
    }
}
#[allow(dead_code)]
impl AddAssign for Co {
    fn add_assign(&mut self, other: Self) {
        *self = self.add(other);
    }
}
#[allow(dead_code)]
impl Add for Co {
    type Output = Self;
    fn add(self, other: Self) -> Self {
        Co::from((self.0).add(other.0))
    }
}
#[allow(dead_code)]
impl SubAssign for Co {
    fn sub_assign(&mut self, other: Self) {
        *self = self.sub(other);
    }
}
#[allow(dead_code)]
impl Sub for Co {
    type Output = Self;
    fn sub(self, other: Self) -> Self {
        Co::from((self.0).sub(other.0))
    }
}
#[allow(dead_code)]
impl MulAssign for Co {
    fn mul_assign(&mut self, other: Self) {
        *self = self.mul(other);
    }
}
#[allow(dead_code)]
impl Mul for Co {
    type Output = Self;
    fn mul(self, other: Self) -> Self {
        Co::from((self.0).mul(other.0))
    }
}
#[allow(dead_code)]
impl DivAssign for Co {
    fn div_assign(&mut self, other: Self) {
        *self = self.div(other);
    }
}
#[allow(dead_code)]
impl Div for Co {
    type Output = Self;
    fn div(self, rhs: Self) -> Self {
        self * rhs.pow((M - 2) as i64)
    }
}
// endregion

// region template
#[allow(dead_code)]
fn iu(i: i64) -> usize { i as usize }

#[allow(dead_code)]
fn ui(i: usize) -> i64 { i as i64 }

#[allow(dead_code, deprecated)]
fn join<T: std::fmt::Display>(slice: &[T], sep: &str) -> String {
    let strings = slice.iter().map(|t| format!("{}", t)).collect::<Vec<_>>();
    strings.connect(sep)
}

#[allow(dead_code)]
fn arr<'a, S, T>(n: usize, mut f: S) -> Vec<T> where S: FnMut(usize) -> T + 'a {
    (0..n).map(|i| f(i)).collect::<Vec<T>>()
}

struct Scanner<'a> {
    cin: StdinLock<'a>,
}

#[allow(dead_code)]
impl<'a> Scanner<'a> {
    fn new(cin: StdinLock<'a>) -> Scanner<'a> {
        Scanner { cin: cin }
    }

    fn read1<T: FromStr>(&mut self) -> Option<T> {
        let token = self.cin.by_ref().bytes().map(|c| c.unwrap() as char)
            .skip_while(|c| c.is_whitespace())
            .take_while(|c| !c.is_whitespace())
            .collect::<String>();
        token.parse::<T>().ok()
    }

    fn read<T: FromStr>(&mut self) -> T {
        self.read1().unwrap()
    }

    fn readn<T: FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read::<T>()).collect()
    }

    fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}
// endregion