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)

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}

fn read_touple<T: std::str::FromStr + Copy>() -> (T, T) {
    let v: Vec<T> = read_vec();
    assert_eq!(v.len(), 2);
    (v[0], v[1])
}

use std::collections::HashMap;

const MOD: usize = 998244353;
fn main() {
    let n: usize = read();
    let mut d: Vec<usize> = read_vec();

    if d[0] != 0 {
        println!("0");
        return;
    }
    d.sort();
    let v = split_same_element(&d);
    //println!("{:?}", v);
    if let Some(&x) = v.get(&0) {
        if x != 1 {
            println!("0");
            return;
        }
    } else {
        println!("0");
        return;
    }

    let mut ans: usize = 1;
    let m = d[n - 1];
    //println!("{}", m);
    for i in 1..m + 1 {
        if !v.contains_key(&i) {
            println!("0");
            return;
        }
        let a: usize = *v.get(&i).unwrap();
        let b: usize = *v.get(&(i - 1)).unwrap();
        ans *= mod_pow(b, a);
        ans %= MOD;
    }

    println!("{}", ans);
}

fn mod_pow(a: usize, b: usize) -> usize {
    let mut result = 1;
    for _ in 0..b {
        result = result * a % MOD;
    }
    return result;
}

fn split_same_element<T: Eq + Copy + std::hash::Hash>(v: &Vec<T>) -> HashMap<T, usize> {
    let mut res = HashMap::new();
    for &e in v {
        *res.entry(e).or_insert(0) += 1;
    }

    res
}