CF1630E.Expected Components

Given a cyclic array $a$ of size $n$ , where $a_i$ is the value of $a$ in the $i$ -th position, there may be repeated values. Let us define that a permutation of $a$ is equal to another permutation of $a$ if and only if their values are the same for each position $i$ or we can transform them to each other by performing some cyclic rotation. Let us define for a cyclic array $b$ its number of components as the number of connected components in a graph, where the vertices are the positions of $b$ and we add an edge between each pair of adjacent positions of $b$ with equal values (note that in a cyclic array the first and last position are also adjacents).

Find the expected value of components of a permutation of $a$ if we select it equiprobably over the set of all the different permutations of $a$ .

The input consists of multiple test cases. The first line contains a single integer $t$ ( $1 \le t \le 10^5$ ) — the number of test cases. Description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $1 \le n \le 10^6$ ) — the size of the cyclic array $a$ .

The second line of each test case contains $n$ integers, the $i$ -th of them is the value $a_i$ ( $1 \le a_i \le n$ ).

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^6$ .

It is guaranteed that the total number of different permutations of $a$ is not divisible by $M$

For each test case print a single integer — the expected value of components of a permutation of $a$ if we select it equiprobably over the set of all the different permutations of $a$ modulo $998\,244\,353$ .

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

In the first test case there is only $1$ different permutation of $a$ :

• $[1, 1, 1, 1]$ has $1$ component.
• Therefore the expected value of components is $\frac{1}{1} = 1$

In the second test case there are $4$ ways to permute the cyclic array $a$ , but there is only $1$ different permutation of $a$ :

• $[1, 1, 1, 2]$ , $[1, 1, 2, 1]$ , $[1, 2, 1, 1]$ and $[2, 1, 1, 1]$ are the same permutation and have $2$ components.
• Therefore the expected value of components is $\frac{2}{1} = 2$

In the third test case there are $6$ ways to permute the cyclic array $a$ , but there are only $2$ different permutations of $a$ :

• $[1, 1, 2, 2]$ , $[2, 1, 1, 2]$ , $[2, 2, 1, 1]$ and $[1, 2, 2, 1]$ are the same permutation and have $2$ components.
• $[1, 2, 1, 2]$ and $[2, 1, 2, 1]$ are the same permutation and have $4$ components.
• Therefore the expected value of components is $\frac{2+4}{2} = \frac{6}{2} = 3$

In the fourth test case there are $120$ ways to permute the cyclic array $a$ , but there are only $24$ different permutations of $a$ :

• Any permutation of $a$ has $5$ components.
• Therefore the expected value of components is $\frac{24\cdot 5}{24} = \frac{120}{24} = 5$