An interesting new Mahonian permutation statistic

The standard algorithm for generating a random permutation gives rise to an obvious permutation statistic $\text{stat}$ that is readily seen to be Mahonian. We give evidence showing that it is not equal to any previously published statistic. Nor does its joint distribution with the standard Eulerian statistics $\text{des}$ and $\text{exc}$ appear to coincide with any known Euler-Mahonian pair. A general construction of Skandera yields an Eulerian partner $\text{ska}$ such that $(\text{ska},\text{stat})$ is equidistributed with $(\text{des},\text{maj})$. However $\text{ska}$ itself appears not to be a known Eulerian statistic. Several ideas for further research on this topic are listed.