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};
use std::str::FromStr;

// https://qiita.com/tubo28/items/e6076e9040da57368845
fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

#[allow(dead_code)]
fn main() {
    let n = read::<usize>();
    let a = (0..n).map(|_| read::<usize>()).collect::<Vec<_>>();
    use std::collections::HashMap;
    let mut m = HashMap::new();
    let mut up = 0;

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

    for i in 0..n {
        let k = a[i];
        up = std::cmp::max(up, k);
        if let Some(&c) = m.get(&k) {
            m.insert(k, c + 1);
        } else {
            m.insert(k, 1);
        }
    }

    if let Some(&c) = m.get(&0) {
        if c > 1 {
            println!("0");
            return;
        }
    }

    let mut ans = 1 as usize;
    let mut prev = 1 as usize;
    for i in 0..up + 1 {
        if let Some(&c) = m.get(&i) {
            ans *= pow(prev, c);
            ans = ans % 998244353;
            prev = c as usize
        } else {
            println!("0");
            return;
        }
    }
    println!("{}", ans % 998244353);

    fn pow(a: usize, b: usize) -> usize {
        let mut ans = a;
        for _ in 0..b - 1 {
            ans *= a;
            ans %= 998244353;
        }
        ans
    }
}