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 C (GCC 5.4.1) (judge verdict: Accepted)

#include <stdio.h>
#include <stdlib.h>

int main()
{
	int i, N, D[100001];
	scanf("%d", &N);
	for (i = 1; i <= N; i++) scanf("%d", &(D[i]));
	
	int count[100001] = {}, flag = 1;
	for (i = 1; i <= N; i++) count[D[i]]++;
	for (i = 1; count[i] > 0; i++);
	for (; i <= N && count[i] == 0; i++);
	if (D[1] != 0 || count[0] > 1 || i <= N) {
		printf("0\n");
		fflush(stdout);
		return 0;
	}

	int j;
	long long ans = 1;
	for (i = 2; count[i] > 0; i++) {
		for (j = 0; j < count[i]; j++) ans = (ans * count[i-1]) % 998244353;
	}
	printf("%lld\n", ans);
	fflush(stdout);
	return 0;
}