Distribution of the inversion statistic on run-sorted permutations

Let $π=π_1\cdots π_n$ be a permutation. We say that $π$ is $run$-$sorted$ if $π_1=1$ and the entries immediately following the descent positions of $π$ form an increasing sequence. Let $\mathcal{R}_n$ denote the set of run-sorted permutations of length $n$, which has cardinality given by the Bell nu