Emordnilap

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). There are $$$n! = n \cdot (n-1) \cdot (n - 2) \cdot \ldots \cdot 1$$$ different permutations of length $$$n$$$.

Given a permutation $$$p$$$ of $$$n$$$ numbers, we create an array $$$a$$$ consisting of $$$2n$$$ numbers, which is equal to $$$p$$$ concatenated with its reverse. We then define the beauty of $$$p$$$ as the number of inversions in $$$a$$$.

The number of inversions in the array $$$a$$$ is the number of pairs of indices $$$i$$$, $$$j$$$ such that $$$i < j$$$ and $$$a_i > a_j$$$.

For example, for permutation $$$p = [1, 2]$$$, $$$a$$$ would be $$$[1, 2, 2, 1]$$$. The inversions in $$$a$$$ are $$$(2, 4)$$$ and $$$(3, 4)$$$ (assuming 1-based indexing). Hence, the beauty of $$$p$$$ is $$$2$$$.

Your task is to find the sum of beauties of all $$$n!$$$ permutations of size $$$n$$$. Print the remainder we get when dividing this value by $$$1\,000\,000\,007$$$ ($$$10^9 + 7$$$).