Given a sequence $a$ consisting of $n$ integers, an inversion is defined as a pair of indices $(i, j)$ such that $i < j$ and $a[i] > a[j]$. Let ${\mathrm{inv}}(a)$ denote the total number of inversions in $a$. A permutation $p$ of order $n$ is a sequence $p$ of size $n$ such that for any $1\le i \le n$, there exists a valid index $j$ such that $p_j = i$. For example, the sequence $p = (1, 3, 2, 4)$ is a permutation of order $4$, and it has 1 inversion, namely $(2, 3)$.

Calculate the sum of ${\mathrm{inv}} (p)$ over all distinct permutations of order $n = 15$. (Two permutations $p$ and $q$ are different if there is an index $i$ for which $p[i] \ne q[i]$.)