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 compare_int(const void *a,const void *b){
	return *(int*)a-*(int*)b;
}

long long rui(int a,int b){
	long long r=1;
	for(int i=0;i<b;i++)r=(r*a)%998244353;
	return r;
}

int main(){
	int n,d[100000],i,c=0,dd[100000]={0};
	long long ans=1;
	scanf("%d",&n);
	for(i=0;i<n;i++)scanf("%d",&d[i]);
	if(d[0]!=0){
		puts("0");
		return 0;
	}
	qsort(d,n,sizeof(int),compare_int);
	if(d[1]==0){
		puts("0");
		return 0;
	}
	for(i=0;i<n;i++)dd[d[i]]++;
	for(i=1;i<=d[n-1];i++)ans=(ans*rui(dd[i-1],dd[i]))%998244353;
	printf("%lld\n",ans);
	return 0;
}