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 (1.42.0) (judge verdict: Accepted)

use proconio::input;
use std::collections::BTreeMap;

const MOD: usize = 998_244_353;

fn main() {
    input! {
        n: usize,
        dd: [usize; n],
    }

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

    let mut map = BTreeMap::new();
    for d in dd {
        *map.entry(d).or_insert(0) += 1;
    }

    let mut ans = 1;
    let mut prev = 1;
    let mut index = 0;
    for (key, value) in map.into_iter() {
        if key == 0 {
            if value != 1 {
                println!("0");
                return;
            }
            index += 1;
            continue;
        }
        if key != index {
            println!("0");
            return;
        }

        for _ in 0..value {
            ans *= prev;
            ans %= MOD;
        }
        index += 1;
        prev = value;
    }

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