Stairs

10000ms

1048576K

Description:

For a permutation $$$p$$$ of numbers $$$1$$$ through $$$n$$$, we define a stair array $$$a$$$ as follows: $$$a_i$$$ is length of the longest segment of permutation which contains position $$$i$$$ and is made of consecutive values in sorted order: $$$[x, x+1, \ldots, y-1, y]$$$ or $$$[y, y-1, \ldots, x+1, x]$$$ for some $$$x \leq y$$$. For example, for permutation $$$p = [4, 1, 2, 3, 7, 6, 5]$$$ we have $$$a = [1, 3, 3, 3, 3, 3, 3]$$$.

You are given the stair array $$$a$$$. Your task is to calculate the number of permutations which have stair array equal to $$$a$$$. Since the number can be big, compute it modulo $$$998\,244\,353$$$. Note that this number can be equal to zero.

Input:

The first line of input contains integer $$$n$$$ ($$$1 \le n \le 10^5$$$) — the length of a stair array $$$a$$$.

The second line of input contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$).

Output:

Print the number of permutations which have stair array equal to $$$a$$$. Since the number can be big, compute it modulo $$$998\,244\,353$$$.

Sample Input:

6
3 3 3 1 1 1

Sample Output:

Sample Input:

7
4 4 4 4 3 3 3

Sample Output:

Sample Input:

1
1

Sample Output:

Sample Input:

8
2 2 2 2 2 2 1 1

Sample Output:

Sample Input:

4
3 2 3 1

Sample Output:

Informação

Provedor Codeforces

Código CF1553I