  Stirling permutations were introduced by Gessel and Stanley, who used their
enumeration by the number of descents to give a combinatorial interpretation of
certain polynomials related to Stirling numbers.
  Quasi-Stirling permutations, which can be viewed as labeled noncrossing
matchings, were introduced by Archer et al. as a natural extension of Stirling
permutations. Janson's correspondence between Stirling permutations and labeled
increasing plane trees extends to a bijection between quasi-Stirling
permutations and the same set of trees without the increasing restriction.
  Archer et al. posed the problem of enumerating quasi-Stirling permutations by
the number of descents, and conjectured that there are $(n+1)^{n-1}$ such
permutations of size $n$ having the maximum number of descents. In this paper
we prove their conjecture, and we give the generating function for
quasi-Stirling permutations by the number of descents, expressed as a
compositional inverse of the generating function of Eulerian polynomials. We
also find the analogue for quasi-Stirling permutations of the main result from
Gessel and Stanley's paper. We prove that the distribution of descents on these
permutations is asymptotically normal, and that the roots of the corresponding
quasi-Stirling polynomials are all real, in analogy to B\'ona's results for
Stirling permutations.
  Finally, we generalize our results to a one-parameter family of permutations
that extends $k$-Stirling permutations, and we refine them by also keeping
track of the number of ascents and the number of plateaus.
