CF1753C.Wish I Knew How to Sort

You are given a binary array $a$ (all elements of the array are $0$ or $1$ ) of length $n$ . You wish to sort this array, but unfortunately, your algorithms teacher forgot to teach you sorting algorithms. You perform the following operations until $a$ is sorted:

1. Choose two random indices $i$ and $j$ such that $i < j$ . Indices are chosen equally probable among all pairs of indices $(i, j)$ such that $1 \le i < j \le n$ .
2. If $a_i > a_j$ , then swap elements $a_i$ and $a_j$ .

What is the expected number of such operations you will perform before the array becomes sorted?

It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{998\,244\,353}$ . Output the integer equal to $p \cdot q^{-1} \bmod 998\,244\,353$ . In other words, output such an integer $x$ that $0 \le x < 998\,244\,353$ and $x \cdot q \equiv p \pmod{998\,244\,353}$ .

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 an integer $n$ ( $1 \le n \le 200\,000$ ) — the number of elements in the binary array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $a_i \in \{0, 1\}$ ) — elements of the array.

It's guaranteed that sum of $n$ over all test cases does not exceed $200\,000$ .

For each test case print one integer — the value $p \cdot q^{-1} \bmod 998\,244\,353$ .

Consider the first test case. If the pair of indices $(2, 3)$ will be chosen, these elements will be swapped and array will become sorted. Otherwise, if one of pairs $(1, 2)$ or $(1, 3)$ will be selected, nothing will happen. So, the probability that the array will become sorted after one operation is $\frac{1}{3}$ , the probability that the array will become sorted after two operations is $\frac{2}{3} \cdot \frac{1}{3}$ , the probability that the array will become sorted after three operations is $\frac{2}{3} \cdot \frac{2}{3} \cdot \frac{1}{3}$ and so on. The expected number of operations is $\sum \limits_{i=1}^{\infty} \left(\frac{2}{3} \right)^{i - 1} \cdot \frac{1}{3} \cdot i = 3$ .

In the second test case the array is already sorted so the expected number of operations is zero.

In the third test case the expected number of operations equals to $\frac{75}{4}$ so the answer is $75 \cdot 4^{-1} \equiv 249\,561\,107 \pmod {998\,244\,353}$ .