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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.regex.Pattern;

public class Main {
    public void solve(BufferedReader stdin, PrintWriter stdout) throws NumberFormatException, IOException {
        int n = Integer.parseInt(stdin.readLine());
        int[] d = toIntArray(stdin.readLine());

        long[] c = new long[Arrays.stream(d).max().getAsInt() + 1];
        for (int v : d) {
            c[v]++;
        }

        if (d[0] != 0 || c[0] != 1) {
            stdout.println(0);
            return ;
        }

        long ans = 1;
        for (int k = 0; k < c.length - 1; k++) {
            ans *= modPow(c[k], c[k + 1]);
            ans %= mod;
        }
        stdout.println(ans);
    }

    private long mod = 998244353;

    private long modPow(long x, long y) {
        long z = 1;
        while (y > 0) {
            if (y % 2 == 0) {
                x = (x * x) % mod;
                y /= 2;
            } else {
                z = (z * x) % mod;
                y--;
            }
        }
        return z;
    }

    private Pattern space = Pattern.compile(" ");

    private int[] toIntArray(String line) {
        return space.splitAsStream(line).mapToInt(Integer::parseInt).toArray();
    }

    public static void main(String[] args) throws NumberFormatException, IOException {
        BufferedReader stdin = new BufferedReader(new InputStreamReader(System.in));
        PrintWriter stdout = new PrintWriter(System.out, false);
        new Main().solve(stdin, stdout);
        stdout.flush();
    }
}