Number Of Permutations

2000ms

262144K

Description:

You are given a sequence of $$$n$$$ pairs of integers: $$$(a_1, b_1), (a_2, b_2), \dots , (a_n, b_n)$$$. This sequence is called bad if it is sorted in non-descending order by first elements or if it is sorted in non-descending order by second elements. Otherwise the sequence is good. There are examples of good and bad sequences:

$$$s = [(1, 2), (3, 2), (3, 1)]$$$ is bad because the sequence of first elements is sorted: $$$[1, 3, 3]$$$;
$$$s = [(1, 2), (3, 2), (1, 2)]$$$ is bad because the sequence of second elements is sorted: $$$[2, 2, 2]$$$;
$$$s = [(1, 1), (2, 2), (3, 3)]$$$ is bad because both sequences (the sequence of first elements and the sequence of second elements) are sorted;
$$$s = [(1, 3), (3, 3), (2, 2)]$$$ is good because neither the sequence of first elements $$$([1, 3, 2])$$$ nor the sequence of second elements $$$([3, 3, 2])$$$ is sorted.

Calculate the number of permutations of size $$$n$$$ such that after applying this permutation to the sequence $$$s$$$ it turns into a good sequence.

A permutation $$$p$$$ of size $$$n$$$ is a sequence $$$p_1, p_2, \dots , p_n$$$ consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ ($$$1 \le p_i \le n$$$). If you apply permutation $$$p_1, p_2, \dots , p_n$$$ to the sequence $$$s_1, s_2, \dots , s_n$$$ you get the sequence $$$s_{p_1}, s_{p_2}, \dots , s_{p_n}$$$. For example, if $$$s = [(1, 2), (1, 3), (2, 3)]$$$ and $$$p = [2, 3, 1]$$$ then $$$s$$$ turns into $$$[(1, 3), (2, 3), (1, 2)]$$$.

Input:

The first line contains one integer $$$n$$$ ($$$1 \le n \le 3 \cdot 10^5$$$).

The next $$$n$$$ lines contains description of sequence $$$s$$$. The $$$i$$$-th line contains two integers $$$a_i$$$ and $$$b_i$$$ ($$$1 \le a_i, b_i \le n$$$) — the first and second elements of $$$i$$$-th pair in the sequence.

The sequence $$$s$$$ may contain equal elements.

Output:

Print the number of permutations of size $$$n$$$ such that after applying this permutation to the sequence $$$s$$$ it turns into a good sequence. Print the answer modulo $$$998244353$$$ (a prime number).

Sample Input:

Sample Output:

Sample Input:

Sample Output:

Sample Input:

Sample Output:

Note:

In first test case there are six permutations of size $$$3$$$:

if $$$p = [1, 2, 3]$$$, then $$$s = [(1, 1), (2, 2), (3, 1)]$$$ — bad sequence (sorted by first elements);
if $$$p = [1, 3, 2]$$$, then $$$s = [(1, 1), (3, 1), (2, 2)]$$$ — bad sequence (sorted by second elements);
if $$$p = [2, 1, 3]$$$, then $$$s = [(2, 2), (1, 1), (3, 1)]$$$ — good sequence;
if $$$p = [2, 3, 1]$$$, then $$$s = [(2, 2), (3, 1), (1, 1)]$$$ — good sequence;
if $$$p = [3, 1, 2]$$$, then $$$s = [(3, 1), (1, 1), (2, 2)]$$$ — bad sequence (sorted by second elements);
if $$$p = [3, 2, 1]$$$, then $$$s = [(3, 1), (2, 2), (1, 1)]$$$ — good sequence.

Informação

Provedor Codeforces

Código CF1207D