Expected Components

2000ms

262144K

Description:

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 $$$a$$$.

Input:

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 $$$M$$$

Output:

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 $$$x \cdot q \equiv p \pmod{M}$$$.

Sample Input:

5
4
1 1 1 1
4
1 1 2 1
4
1 2 1 2
5
4 3 2 5 1
12
1 3 2 3 2 1 3 3 1 3 3 2

Sample Output:

Note:

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 $$$\frac{24\cdot 5}{24} = \frac{120}{24} = 5$$$

Informação

Provedor Codeforces

Código CF1630E