Wish I Knew How to Sort
Description
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:
- 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 $1 \le i < j \le n$.
- 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 $p \cdot q^{-1} \bmod 998\,244\,353$.
Input
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 $200\,000$.
Output
For each test case print one integer — the value $p \cdot q^{-1} \bmod 998\,244\,353$.
3
3
0 1 0
5
0 0 1 1 1
6
1 1 1 0 0 1
3
0
249561107
Note
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 $75 \cdot 4^{-1} \equiv 249\,561\,107 \pmod {998\,244\,353}$.