311355: CF1973E. Cat, Fox and Swaps
Description
Fox has found an array $p_1, p_2, \ldots, p_n$, that is a permutation of length $n^\dagger$ of the numbers $1, 2, \ldots, n$. She wants to sort the elements in increasing order. Cat wants to help her — he is able to swap any two numbers $x$ and $y$ in the array, but only if $l \leq x + y \leq r$ (note that the constraint is imposed on the values of the elements, not their positions). He can make such swaps any number of times.
They don't know the numbers $l$, $r$ yet, they only know that it's true that $1 \leq l \leq r \leq 2 \cdot n$.
You are given the number $n$ and the array $p_1, p_2, \ldots, p_n$. Determine how many pairs $(l, r)$ satisfying the conditions are there such that you can sort the permutation if you can only swap two number $(x, y)$ such that $l \leq x + y \leq r$ (arbitrary number of times, possibly $0$).
$^\dagger$ A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array).
InputEach test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.
Description of each test case consists of two lines. The first line contains one integer $n$ ($1 \leq n \leq 10^5$).
The second line contains $n$ integers: the array $p_1, p_2, \ldots, p_n$ ($1 \le p_i \le n$). It is guaranteed that this array is a permutation of length $n$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
OutputFor each test case, print the number of pairs of integers $(l, r)$ such that $1 \leq l \leq r \leq 2 \cdot n$, and you can sort the array under the constraints.
ExampleInput7 2 2 1 3 3 1 2 4 3 2 1 4 5 5 3 1 2 4 5 1 2 3 4 5 6 3 2 1 5 4 6 6 1 3 2 4 5 6Output
6 11 23 29 55 46 58Note
In the first example, we need to be able to swap $1$ and $2$, so we must be able to swap numbers with sum $3$. There are exactly $6$ pairs satisfying the condition: $(1, 3), (2, 3), (3, 3), (1, 4), (2, 4)$ and $(3, 4)$, so the answer is $6$.
In the second example, the $11$ pairs satisfying the condition are $(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 5)$ and $(4, 6)$. For example, if we pick the pair $(3, 4)$ we can first swap the numbers $1$ and $2$ and then the numbers $1$ and $3$, after this, the permutation is sorted.
Output
狐狸发现了一个长度为n的数组p_1, p_2, ..., p_n，该数组是数字1, 2, ..., n的一个排列。她想要将数组元素按升序排序。猫想要帮助她——他能够交换数组中的任意两个数字x和y，但只有当l <= x + y <= r时才行（注意，这个限制是针对元素的值，而不是它们的位置）。他可以进行任意次数的这种交换。
他们还不知道数字l, r是多少，他们只知道1 <= l <= r <= 2 * n。
给你数字n和数组p_1, p_2, ..., p_n。确定有多少对(l, r)满足条件，即你只能交换两个数字(x, y)，使得l <= x + y <= r（任意次数，可能为0），这样就能排序排列。
输入数据格式：
每个测试包含多个测试用例。第一行包含测试用例的数量t (1 <= t <= 10^4)。接下来是每个测试用例的描述。
每个测试用例包含两行。第一行包含一个整数n (1 <= n <= 10^5)。
第二行包含n个整数：数组p_1, p_2, ..., p_n (1 <= p_i <= n)。保证这个数组是一个长度为n的排列。
保证所有测试用例的n之和不超过10^5。
输出数据格式：
对于每个测试用例，打印满足1 <= l <= r <= 2 * n的整数对(l, r)的数量，使得你可以在限制条件下排序数组。题目大意： 狐狸发现了一个长度为n的数组p_1, p_2, ..., p_n，该数组是数字1, 2, ..., n的一个排列。她想要将数组元素按升序排序。猫想要帮助她——他能够交换数组中的任意两个数字x和y，但只有当l <= x + y <= r时才行（注意，这个限制是针对元素的值，而不是它们的位置）。他可以进行任意次数的这种交换。 他们还不知道数字l, r是多少，他们只知道1 <= l <= r <= 2 * n。 给你数字n和数组p_1, p_2, ..., p_n。确定有多少对(l, r)满足条件，即你只能交换两个数字(x, y)，使得l <= x + y <= r（任意次数，可能为0），这样就能排序排列。 输入数据格式： 每个测试包含多个测试用例。第一行包含测试用例的数量t (1 <= t <= 10^4)。接下来是每个测试用例的描述。 每个测试用例包含两行。第一行包含一个整数n (1 <= n <= 10^5)。 第二行包含n个整数：数组p_1, p_2, ..., p_n (1 <= p_i <= n)。保证这个数组是一个长度为n的排列。 保证所有测试用例的n之和不超过10^5。 输出数据格式： 对于每个测试用例，打印满足1 <= l <= r <= 2 * n的整数对(l, r)的数量，使得你可以在限制条件下排序数组。