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.util.Scanner;

public class Main {
    public static void debug(String str) {
        System.out.println(str);
    }

    public static void main(String[] args) {

        int N;
        long[] Di;
        long E = 998244353;

        try (Scanner sc = new Scanner(System.in)) {
            N = sc.nextInt();
            Di = new long[N];
            for (int i = 0; i < Di.length; i++) {
                Di[i] = sc.nextLong();
            }
        }
        if (N == 1) {
            System.out.println(1);
            return;
        }
        if (Di[0] != 0) {
            System.out.println(0);
            return;
        }
        long[] Num = new long[N];
        long maxDis = 0;
        for (long l : Di) {
            Num[(int) l]++;
            if (l >= maxDis) {
                maxDis = l;
            }
        }
        //System.out.println(Arrays.toString(Num));
        //System.out.println(maxDis);
        for (int i = 0; i <= maxDis; i++) {
            if (Num[i] == 0) {
                System.out.println(0);
                return;
            }
        }
        if (Num[0] != 1) {
            System.out.println(0);
            return;
        }
        long result = 1;
        for (int i = 1; i <= maxDis; i++) {
            long up = Num[i - 1];
            long down = Num[i];
            long temp = 1;
            up = up % E;
            while (down > 0) {
                temp = temp * up;
                temp = temp % E;
                down--;
            }
            result = result * temp;
            result = result % E;
        }
        System.out.println(result);
    }
}