A14369 | Top-Notch Insertions

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 $998\,244\,353$ .