  Given a permutation statistic $\operatorname{st}$, define its inverse
statistic $\operatorname{ist}$ by
$\operatorname{ist}(\pi):=\operatorname{st}(\pi^{-1})$. We give a general
approach, based on the theory of symmetric functions, for finding the joint
distribution of $\operatorname{st}_{1}$ and $\operatorname{ist}_{2}$ whenever
$\operatorname{st}_{1}$ and $\operatorname{st}_{2}$ are descent statistics:
permutation statistics that depend only on the descent composition. We apply
this method to a number of descent statistics, including the descent number,
the peak number, the left peak number, the number of up-down runs, and the
major index. Perhaps surprisingly, in many cases the polynomial giving the
joint distribution of $\operatorname{st}_{1}$ and $\operatorname{ist}_{2}$ can
be expressed as a simple sum involving products of the polynomials giving the
(individual) distributions of $\operatorname{st}_{1}$ and
$\operatorname{st}_{2}$. Our work leads to a rederivation of Stanley's
generating function for doubly alternating permutations, as well as several
conjectures concerning real-rootedness and $\gamma$-positivity.
