  We exploit Krattenthaler's bijection between 123-avoiding permutations and
Dyck paths to determine the Eulerian distribution over the set $S_n(123)$ of
123-avoiding permutations in $S_n$. In particular, we show that the descents of
a permutation correspond to valleys and triple falls of the associated Dyck
path. We get the Eulerian numbers of $S_n(123)$ by studying the joint
distribution of these two statistics on Dyck paths.
