Top-Notch Insertions

对于一个长度为 $n$ 的序列 $a$ 做插入排序，即依次考虑 $a_2, \cdots, a_n$ 每个元素： * 如果 $a_i \ge a_{i - 1}$，即前缀依然保持不下降，不做操作。 * 否则，找到最靠前的位置 $p$，使得 $a_i < a_p$，然后将 $a_i$ 插入到 $a_p$ 前面，并重新标号序列。记这次插入为 $(i, p)$。 接下来给定 $m$ 个二元组 $(x_i, y_i)$，求有多少个长度为 $n$ 的序列 $a$，满足 $\forall i, a_i \in [1, n] \cap \mathbb N$ 且做插入排序恰好进行给定的 $m$ 次插入。 输出序列的数量对 $998244353$ 取模的结果。

Consider the insertion sort algorithm used to sort an integer sequence $ [a_1, a_2, \ldots, a_n] $ of length $ n $ in non-decreasing order. For each $ i $ in order from $ 2 $ to $ n $ , do the following. If $ a_i \ge a_{i-1} $ , do nothing and move on to the next value of $ i $ . Otherwise, find the smallest $ j $ such that $ a_i < a_j $ , shift the elements on positions from $ j $ to $ i-1 $ by one position to the right, and write down the initial value of $ a_i $ to position $ j $ . In this case we'll say that we performed an insertion of an element from position $ i $ to position $ j $ . It can be noticed that after processing any $ i $ , the prefix of the sequence $ [a_1, a_2, \ldots, a_i] $ is sorted in non-decreasing order, therefore, the algorithm indeed sorts any sequence. For example, sorting $ [4, 5, 3, 1, 3] $ proceeds as follows: - $ i = 2 $ : $ a_2 \ge a_1 $ , do nothing; - $ i = 3 $ : $ j = 1 $ , insert from position $ 3 $ to position $ 1 $ : $ [3, 4, 5, 1, 3] $ ; - $ i = 4 $ : $ j = 1 $ , insert from position $ 4 $ to position $ 1 $ : $ [1, 3, 4, 5, 3] $ ; - $ i = 5 $ : $ j = 3 $ , insert from position $ 5 $ to position $ 3 $ : $ [1, 3, 3, 4, 5] $ . You are given an integer $ n $ and a list of $ m $ integer pairs $ (x_i, y_i) $ . We are interested in sequences such that if you sort them using the above algorithm, exactly $ m $ insertions will be performed: first from position $ x_1 $ to position $ y_1 $ , then from position $ x_2 $ to position $ y_2 $ , ..., finally, from position $ x_m $ to position $ y_m $ . How many sequences of length $ n $ consisting of (not necessarily distinct) integers between $ 1 $ and $ n $ , inclusive, satisfy the above condition? Print this number modulo $ 998\,244\,353 $ .

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10^5 $ ). Description of the test cases follows. The first line of each test case contains two integers $ n $ and $ m $ ( $ 2 \le n \le 2 \cdot 10^5 $ ; $ 0 \le m < n $ ) — the length of the sequence and the number of insertions. The $ i $ -th of the following $ m $ lines contains two integers $ x_i $ and $ y_i $ ( $ 2 \le x_1 < x_2 < \ldots < x_m \le n $ ; $ 1 \le y_i < x_i $ ). These lines describe the sequence of insertions in chronological order. It is guaranteed that the sum of $ m $ over all test cases does not exceed $ 2 \cdot 10^5 $ . Note that there is no constraint on the sum of $ n $ of the same kind.

For each test case, print the number of sequences of length $ n $ consisting of integers from $ 1 $ to $ n $ such that sorting them with the described algorithm produces the given sequence of insertions, modulo $ 998\,244\,353 $ .

3
3 0
3 2
2 1
3 1
5 3
3 1
4 1
5 3

10
1
21

In the first test case, the algorithm performs no insertions — therefore, the initial sequence is already sorted in non-decreasing order. There are $ 10 $ such sequences: $ [1, 1, 1], [1, 1, 2], [1, 1, 3], [1, 2, 2], [1, 2, 3], [1, 3, 3], [2, 2, 2], [2, 2, 3], [2, 3, 3], [3, 3, 3] $ . In the second test case, the only sequence satisfying the conditions is $ [3, 2, 1] $ . In the third test case, $ [4, 5, 3, 1, 3] $ is one of the sought sequences.