  Motivated by recent results on quasi-Stirling permutations, which are
permutations of the multiset $\{1,1,2,2,\dots,n,n\}$ that avoid the "crossing"
patterns 1212 and 2121, we consider nonnesting permutations, defined as those
that avoid the patterns 1221 and 2112 instead. We show that the polynomial
giving the distribution of the number of descents on nonnesting permutations is
a product of an Eulerian polynomial and a Narayana polynomial. It follows that,
rather unexpectedly, this polynomial is palindromic. We provide bijective
proofs of these facts by composing various transformations on Dyck paths,
including the Lalanne--Kreweras involution.
