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)

const MOD: usize = 998244353;
fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner(s.lock());
    let n: usize = sc.read();
    let d: Vec<usize> = sc.vec(n);
    if d[0] != 0 {
        println!("0");
        return;
    }
    let mut count = vec![0; n];
    for d in d.into_iter() {
        count[d] += 1;
    }
    if count[0] != 1 {
        println!("0");
        return;
    }

    while let Some(tail) = count.pop() {
        if tail != 0 {
            count.push(tail);
            break;
        }
    }

    let mut ans = 1;
    for i in 1..count.len() {
        let prev = count[i - 1];
        let next = count[i];
        let mut p = 1;
        for _ in 0..next {
            p = (p * prev) % MOD;
        }
        ans *= p;
        ans %= MOD;
    }
    println!("{}", ans);
}

pub struct Scanner<R>(R);

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .0
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n')
            .take_while(|&b| b != b' ' && b != b'\n')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}