Bermutation

You are given a permutation $p$ of integers from $1$ to $n$. You may apply the following operation any number of times. Choose a segment of $2b$ consecutive elements and swap its two halves. Formally, if you choose the segment $a_i, a_{i+1}, \ldots, a_{i+2b-1}$, the result is $a_{i+b}, a_{i+b+1}, \ldots, a_{i+2b-1}, a_i, a_{i+1}, \ldots, a_{i+b-1}$.

Let $S$ be the set of all permutations that can be obtained from $p$ by applying this operation zero or more times. Each operation chooses its segment independently of the others. List all permutations in $S$ in lexicographical order, numbering them from $1$. Find the position of $p$ in this list. Print the answer modulo $120\,586\,241$.

The first line contains one integer $T$, the number of test cases. Each test case is given on two lines. The first line contains two integers $n$ and $b$ ($2 \le n \le 10^5$, $1 \le b$, $2b \le n$). The second line contains the $n$ elements of the permutation $p$. Each integer from $1$ to $n$ appears exactly once.

The sum of $n$ over all test cases does not exceed $10^5$.

For each test case, print the 1-based position of $p$ in the lexicographically ordered list of all permutations obtainable from $p$, modulo $120\,586\,241$.