D. Number Of Permutations
combinatorics
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 \([(1, 3), (2, 3), (1, 2)]\).
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.
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).
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.