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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
#define mod 998244353

long long modpow(int n, int r) {
    long long ret = 1; long long tmp = (long long) n;
    while (r != 0) {
        if (r % 2) ret *= tmp;
        tmp *= tmp; tmp %= mod; ret %= mod;
        r /= 2;
    }
    return ret;
}

int main() {
    int n; cin >> n;
    vector<int> d(n);
    for (int i = 0; i < n; i++) cin >> d[i];
    map<int, int> m;
    for (int i = 0; i < n; i++) m[d[i]]++;
    if (d[0] != 0 || m[0] != 1) {
        cout << 0 << endl;
        return 0;
    }
    long long ans = 1;
    for (int i = 1; i < n; i++) {
        if (m[i] == 0 && m[i-1] == 0) continue;
        ans = ans * modpow(m[i-1], m[i]) % mod;
    }
    cout << ans << endl;
    return 0;
}