Inversion of a permutation ($\mathrm{inv}(\pi )$)

For a permutation $\pi$ of $\{1,\dots ,n\}$, an inversion is a pair $i<j$ with $\pi (i)>\pi (j)$. Its number equals the Coxeter length for adjacent transpositions: each adjacent swap changes the count by one, and swapping adjacent descents sorts the permutation in exactly that many steps.